Classical Mechanics, Forces, and Energy Transformations

Kinematics: Position, Distance, and Displacement

Kinematics is the part of physics that describes motion. Before we can talk about how fast something moves or how its motion changes, we need to describe where it is and how it moves from one place to another.

Three important ideas in kinematics are position, distance, and displacement. These words sound similar, but they do not mean the same thing. Understanding the difference helps you read motion problems correctly and solve them accurately.

In this lesson, you will learn how to describe an object's location, how to measure the total path it travels, and how to find its overall change in position.

1. Position: Where an object is

Position tells the location of an object compared to a chosen starting point called a reference point or origin.

For example, imagine a straight sidewalk. If we choose a lamp post as the origin, then a student standing 4 meters to the right of the lamp post has a position of +4 m. A student standing 3 meters to the left has a position of -3 m.

On a number line, positions to the right are usually positive, and positions to the left are usually negative.

We often use the symbol \(x\) for position on a straight line. For example:

means the object is 6 meters to the right of the origin.

Important idea: Position is not about how far the object traveled. It is only about where the object is now.

2. Distance: Total path traveled

Distance is the total length of the path an object travels.

Distance only cares about how much ground was covered. It does not include direction. Because of this, distance is a scalar quantity.

A scalar has magnitude only. That means it tells "how much" but not "which way."

For example, if you walk 5 meters forward and then 2 meters backward, your total distance is:

Even though you turned around, distance adds up the whole path.

3. Displacement: Change in position

Displacement tells how far and in what direction an object's position changes from start to finish.

Displacement compares the final position to the initial position. It does not depend on the full path taken. Because displacement includes direction, it is a vector quantity.

A vector has both magnitude and direction.

The formula for displacement is:

Using symbols:

where:

• \(\Delta x\) = displacement
• \(x_f\) = final position
• \(x_i\) = initial position

If the result is positive, the displacement is in the positive direction. If the result is negative, the displacement is in the negative direction.

4. Distance vs. displacement

These two ideas are often confused, so it helps to compare them directly.

• Distance is the total path traveled.
• Displacement is the straight-line change from start to finish.
• Distance has no direction.
• Displacement includes direction.
• Distance is always zero or positive.
• Displacement can be positive, negative, or zero.

An object can travel a large distance but have a small displacement.

For example, if you walk around the school track once and end where you started, your distance is the length of the whole track, but your displacement is zero because your starting and ending positions are the same.

5. Choosing an origin and direction

To describe position and displacement clearly, you must choose:

• an origin or reference point
• a positive direction

For example, on a straight road, you might choose the mailbox as \(0\text{ m}\), east as positive, and west as negative.

Then a car at \(+20\text{ m}\) is 20 meters east of the mailbox. A car at \(-15\text{ m}\) is 15 meters west of the mailbox.

This is why position and displacement often use positive and negative numbers.

6. Worked Example 1: Finding position

A dog is sitting 8 meters to the right of a tree. The tree is chosen as the origin. What is the dog's position?

Step 1: Identify the origin and direction.

The tree is at \(0\text{ m}\). Right is positive.

Step 2: Write the position.

Answer: The dog's position is \(+8\text{ m}\).

7. Worked Example 2: Distance and displacement on a line

A student walks 10 meters east, then 4 meters west. Find the distance and displacement.

Step 1: Find distance.

Distance is the total path traveled, so we add both parts:

Step 2: Find displacement.

Let east be positive.

The student first goes \(+10\text{ m}\), then \(-4\text{ m}\).

Answer:

• Distance = 14 m
• Displacement = \(+6\text{ m}\) or 6 m east

Notice that the distance is larger than the displacement because the student changed direction.

8. Worked Example 3: Using the displacement formula

A bicycle starts at position \(x_i = -3\text{ m}\) and ends at position \(x_f = +5\text{ m}\). What is its displacement?

Step 1: Use the formula.

Step 2: Substitute the values.

Step 3: Simplify.

Answer: The bicycle's displacement is \(+8\text{ m}\).

This means the bicycle's position changed 8 meters in the positive direction.

9. Worked Example 4: Returning to the starting point

A runner starts at the school gate, runs 50 meters north, then turns around and runs 50 meters south back to the gate.

Find the distance.

Find the displacement.

The runner ends at the starting point, so the change in position is:

Answer:

• Distance = 100 m
• Displacement = 0 m

This is a very important example. An object can move a lot and still have zero displacement if it ends where it started.

10. Common mistakes to avoid

• Mixing up distance and displacement: Distance is total path; displacement is change in position.
• Ignoring direction: Displacement must include direction or a positive/negative sign.
• Adding positions instead of subtracting: Use \(\Delta x = x_f - x_i\).
• Thinking distance can be negative: Distance is never negative.
• Forgetting the reference point: Position only makes sense when you know the origin.

11. Quick check for understanding

1. An object is 12 m left of the origin. What is its position?
2. A person walks 7 m south, then 3 m south. What is the distance?
3. A person walks 7 m south, then 3 m south. What is the displacement?
4. A car starts at \(+2\text{ m}\) and ends at \(-4\text{ m}\). What is its displacement?

Answers:

1. \(-12\text{ m}\)
2. \(10\text{ m}\)
3. \(10\text{ m}\) south
4. $$\Delta x = -4 - 2 = -6\text{ m}$$

12. Summary

Position tells where an object is compared to an origin. Distance is the total length of the path traveled and has no direction. Displacement is the change in position from start to finish and includes direction.

When solving motion problems, always ask:

• Where did the object start?
• Where did it end?
• Did it change direction?
• Am I finding total path length or overall change in position?

If you can answer those questions, you can correctly tell the difference between position, distance, and displacement.

Distance, Displacement, Speed, and Velocity are all words we use to describe motion. Motion means how something moves.

These words sound similar, but they do different jobs. When you understand each one, you can describe movement clearly and correctly.

In this lesson, you will learn:

• what distance means
• what displacement means
• what speed means
• what velocity means
• how some motion words need only a number, while others need a number and a direction

Let’s start with the easiest idea: distance.

Distance is how far something travels. It tells the total length of the path.

If you walk 3 steps forward and then 2 more steps forward, your distance is 5 steps. We add all the parts of the trip.

If you walk around a playground, distance tells how much ground you covered, no matter which way you turned.

Distance is a scalar quantity. That means it needs only a number and a unit, like 5 meters or 10 seconds. For distance, we do not need direction.

Now let’s learn displacement.

Displacement tells how far an object is from where it started and in what direction.

This means displacement looks at the starting point and the ending point. It does not care about every twist and turn along the way.

Imagine you walk 4 meters east, then 1 meter west. You traveled a distance of 5 meters total. But you ended up only 3 meters east of where you started. So your displacement is 3 meters east.

Displacement is a vector quantity. That means it needs a number, a unit, and a direction.

Here is a simple way to remember the difference:

• Distance = total path traveled
• Displacement = change from start to finish, with direction

Now let’s talk about speed.

Speed tells how fast something moves. It compares distance to time.

We can find speed with this rule:

$$\text{speed} = \frac{\text{distance}}{\text{time}}$$

If a bike travels 12 meters in 3 seconds, its speed is:

$$\frac{12}{3} = 4$$

So the speed is 4 meters per second.

Speed is also a scalar quantity. It tells only how fast, not which way.

Next is velocity.

Velocity tells both speed and direction. It is like speed, but with direction added.

If a runner moves 6 meters in 2 seconds to the east, the velocity is 3 meters per second east.

Velocity is a vector quantity because direction matters.

Here is another helpful way to remember:

• Speed = how fast
• Velocity = how fast and which way

Let’s compare all four ideas together.

• Distance: how far traveled
• Displacement: how far from the start, with direction
• Speed: distance traveled each amount of time
• Velocity: displacement each amount of time, with direction

Another way to say velocity is:

$$\text{velocity} = \frac{\text{displacement}}{\text{time}}$$

For 4th grade, it is most important to remember that velocity includes direction.

Why does direction matter?

Direction helps us describe motion more exactly. Two objects can have the same speed but different velocity.

For example, one car may move at 20 meters per second east, and another car may move at 20 meters per second west. They have the same speed, but different velocity.

What if you go out and come back?

This is where distance and displacement can be very different.

If you walk 10 meters east and then 10 meters west, your total distance is 20 meters. But you end where you started.

So your displacement is 0 meters. You are not away from your starting point at all.

That also means your trip had distance, but your overall change in position was zero.

Let’s look at some worked examples.

Worked Example 1: Finding Distance

A student walks 2 meters to the door, then 3 more meters to the table. What is the distance traveled?

Add the parts of the trip:

$$2 + 3 = 5$$

The distance is 5 meters.

Worked Example 2: Finding Displacement

A dog runs 7 meters north, then 2 meters south. What is the displacement?

First, think about where the dog ends compared with where it started.

The dog went 7 meters north but came back 2 meters south.

$$7 - 2 = 5$$

The displacement is 5 meters north.

The total distance is different:

$$7 + 2 = 9$$

So:

• Distance = 9 meters
• Displacement = 5 meters north

Worked Example 3: Finding Speed

A toy car travels 15 meters in 5 seconds. What is its speed?

Use the speed rule:

$$\text{speed} = \frac{\text{distance}}{\text{time}}$$

Substitute the numbers:

$$\text{speed} = \frac{15}{5} = 3$$

The speed is 3 meters per second.

Worked Example 4: Comparing Speed and Velocity

A bird flies 8 meters east in 4 seconds. What are its speed and velocity?

First find speed:

$$\text{speed} = \frac{8}{4} = 2$$

The speed is 2 meters per second.

Now include direction for velocity.

The velocity is 2 meters per second east.

Notice that the number is the same, but velocity includes direction.

Important ideas to remember

• Distance and speed are scalars. They do not need direction.
• Displacement and velocity are vectors. They do need direction.
• Distance can never be less than displacement when you compare how far something moved.
• If you return to where you started, your displacement is 0.
• If you know how far and how long, you can find speed.
• If you know how far from start to finish and the direction, you can describe displacement and velocity.

Quick check questions

1. A child walks 6 meters east and then 4 meters west. What is the distance? What is the displacement?
2. A ball rolls 10 meters in 2 seconds. What is its speed?
3. A skateboard moves 12 meters south in 3 seconds. What is its velocity?

Answers

1. Distance: $$6 + 4 = 10$$ so 10 meters. Displacement: $$6 - 4 = 2$$ so 2 meters east.
2. $$\text{speed} = \frac{10}{2} = 5$$ so 5 meters per second.
3. $$\frac{12}{3} = 4$$ so the velocity is 4 meters per second south.

Summary

Distance tells the total path traveled. Displacement tells where you are compared with where you started, and it must include direction.

Speed tells how fast something moves using distance and time. Velocity tells how fast something moves and in what direction.

When you see a motion problem, ask yourself: Do I need only how much, or do I also need direction? That question will help you choose the right word.

Speed, velocity, and relative motion are all ways to describe how objects move. In everyday life, we often say something is “going fast” or “moving slowly,” but in science we use more exact ideas. Understanding these ideas helps us describe motion clearly and solve problems correctly.

This lesson will explain the difference between speed and velocity, show how to calculate average and instantaneous velocity, and introduce relative motion, which explains why motion can look different from different points of view.

Before starting, remember two important motion words:

• Distance: how much ground an object covers.
• Displacement: the change in position from start to finish, including direction.

These two ideas are important because speed uses distance, while velocity uses displacement.

1. What is speed?

Speed tells how fast something moves. It does not include direction. Speed is a scalar, which means it has size but no direction.

The formula for average speed is:

$$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$

Common units for speed are meters per second \((\text{m/s})\), kilometers per hour \((\text{km/h})\), or miles per hour \((\text{mph})\).

For example, if a student walks 100 meters in 20 seconds, the average speed is:

$$\text{speed} = \frac{100\ \text{m}}{20\ \text{s}} = 5\ \text{m/s}$$

This means the student covers 5 meters every second on average.

2. What is velocity?

Velocity tells both how fast an object moves and in what direction. Velocity is a vector, which means it has both size and direction.

The formula for average velocity is:

$$\text{average velocity} = \frac{\text{displacement}}{\text{time}}$$

If direction is included, a complete answer might be something like 5 m/s east or 12 m/s downward.

Here is the key difference:

• Speed uses distance.
• Velocity uses displacement.
• Speed has no direction.
• Velocity includes direction.

3. Distance and displacement are not always the same

If you walk 30 meters east and then 30 meters west, your distance traveled is 60 meters because that is the total path length.

But your displacement is 0 meters because you ended where you started. Your starting and ending positions are the same.

This means an object can have a speed greater than zero but an average velocity of zero.

4. Average speed and average velocity

In real life, objects do not always move at a perfectly steady rate. They may speed up, slow down, or change direction. That is why we often use average values.

Average speed looks at the whole trip:

$$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$

Average velocity also looks at the whole trip, but uses displacement:

$$\text{average velocity} = \frac{\text{change in position}}{\text{total time}}$$

If an object moves in a straight line without changing direction, distance and displacement may be the same. In that case, speed and the magnitude of velocity are the same number.

5. Instantaneous speed and instantaneous velocity

Instantaneous speed means the speed of an object at one specific moment. A car’s speedometer shows instantaneous speed.

Instantaneous velocity means the velocity of an object at one specific moment. It tells both the speed and the direction at that instant.

For example, if a car is turning left at a certain moment and the speedometer reads 18 m/s, then its instantaneous speed is 18 m/s. Its instantaneous velocity is 18 m/s in the direction it is moving at that moment.

So, average values describe motion over a time interval, while instantaneous values describe motion at a single instant.

6. Choosing a direction sign

In many problems, directions are shown with positive and negative signs. For example:

• East or right may be positive.
• West or left may be negative.

If a runner moves 40 meters east, the displacement could be written as \(+40\ \text{m}\). If the runner moves 15 meters west, it could be written as \(-15\ \text{m}\).

Using signs makes it easier to calculate displacement and velocity.

7. What is relative motion?

Relative motion means that the motion of an object depends on the reference frame of the observer. A reference frame is the point of view from which motion is measured.

For example, if you are sitting in a bus moving forward, the person next to you may appear to be at rest. But to someone standing on the sidewalk, both of you are moving with the bus.

This means motion is often described relative to something else.

8. Reference frames

A reference frame is like a background or viewpoint used to measure motion.

• From the ground reference frame, a train may be moving at 20 m/s east.
• From a passenger’s reference frame inside the train, another seated passenger may appear not to move at all.

Neither description is wrong. They are simply from different reference frames.

In this lesson, we focus on simple situations involving inertial reference frames. These are reference frames that are not speeding up, slowing down, or turning. At this level, you can think of them as reference frames moving at constant velocity or staying still.

9. Relative velocity

Relative velocity compares the velocity of one object to another. In simple one-direction problems, relative velocity can often be found by adding or subtracting velocities.

If two objects move in the same direction, subtract their speeds to find how fast one appears to move relative to the other.

If two objects move in opposite directions, add their speeds to find how fast they approach or separate from each other.

In symbols, one useful idea is:

$$v_{A\text{ relative to }B} = v_A - v_B$$

This means the velocity of object A relative to object B equals A’s velocity minus B’s velocity, as long as you use the same positive direction for both.

10. Worked Example 1: Finding average speed

A cyclist travels 150 meters in 30 seconds. What is the cyclist’s average speed?

Step 1: Write the formula.

$$\text{average speed} = \frac{\text{distance}}{\text{time}}$$

Step 2: Substitute the values.

$$\text{average speed} = \frac{150\ \text{m}}{30\ \text{s}}$$

Step 3: Calculate.

$$\text{average speed} = 5\ \text{m/s}$$

Answer: The cyclist’s average speed is 5 m/s.

11. Worked Example 2: Average velocity with direction

A runner moves 80 meters east in 10 seconds. What is the runner’s average velocity?

Step 1: Use the formula.

$$\text{average velocity} = \frac{\text{displacement}}{\text{time}}$$

Step 2: Substitute the values.

$$\text{average velocity} = \frac{80\ \text{m east}}{10\ \text{s}}$$

Step 3: Calculate.

$$\text{average velocity} = 8\ \text{m/s east}$$

Answer: The runner’s average velocity is 8 m/s east.

Notice that the answer includes a direction. Without direction, it would only be speed.

12. Worked Example 3: Speed and velocity are different

A student walks 60 meters north to a library, then walks 60 meters south back to the starting point. The whole trip takes 40 seconds.

Find:

• average speed
• average velocity

Step 1: Find total distance.

The student walks 60 m north and 60 m south.

$$\text{total distance} = 60\ \text{m} + 60\ \text{m} = 120\ \text{m}$$

Step 2: Find average speed.

$$\text{average speed} = \frac{120\ \text{m}}{40\ \text{s}} = 3\ \text{m/s}$$

Step 3: Find displacement.

The student ends at the starting point, so:

$$\text{displacement} = 0\ \text{m}$$

Step 4: Find average velocity.

$$\text{average velocity} = \frac{0\ \text{m}}{40\ \text{s}} = 0\ \text{m/s}$$

Answer:

• Average speed = 3 m/s
• Average velocity = 0 m/s

This example clearly shows why speed and velocity are not the same.

13. Worked Example 4: Relative motion on a train

A train moves east at 20 m/s relative to the ground. A passenger inside the train walks east at 2 m/s relative to the train. What is the passenger’s velocity relative to the ground?

Step 1: Think about the reference frames.

The train is already moving east. The passenger is also walking east inside the train.

Step 2: Since both are in the same direction, add the velocities.

$$v_{\text{passenger, ground}} = 20\ \text{m/s} + 2\ \text{m/s}$$

Step 3: Calculate.

$$v_{\text{passenger, ground}} = 22\ \text{m/s east}$$

Answer: The passenger’s velocity relative to the ground is 22 m/s east.

Now imagine the same passenger walks west at 2 m/s relative to the train while the train still moves east at 20 m/s. Then the passenger’s velocity relative to the ground would be:

$$20\ \text{m/s east} - 2\ \text{m/s} = 18\ \text{m/s east}$$

The passenger is still moving east relative to the ground, but more slowly than the train.

14. Common mistakes to avoid

• Mixing up distance and displacement. Distance is total path length. Displacement is change in position.
• Forgetting direction in velocity. A velocity answer should include direction, such as north, south, east, west, left, or right.
• Using the wrong formula. For speed, use total distance. For velocity, use displacement.
• Ignoring the reference frame. In relative motion, always ask, “Compared to what?”
• Adding when you should subtract. In relative motion, the direction of motion matters.

15. Quick comparison chart

• Speed: how fast something moves
• Velocity: speed with direction
• Average speed: total distance divided by total time
• Average velocity: displacement divided by total time
• Instantaneous speed: speed at one moment
• Instantaneous velocity: velocity at one moment
• Relative motion: motion described from a certain reference frame

16. How to solve motion questions step by step

1. Read the question carefully.
2. Decide whether it asks for speed or velocity.
3. Identify the known values: distance, displacement, time, and direction.
4. Choose the correct formula.
5. Substitute the numbers with units.
6. Calculate carefully.
7. Check whether your final answer needs a direction.
8. If it is a relative motion problem, identify the reference frame.

17. Brief summary

Speed tells how fast an object moves and uses total distance. Velocity tells how fast and in what direction an object moves and uses displacement.

Average speed and velocity describe motion over a time interval, while instantaneous speed and velocity describe motion at a single moment. Relative motion reminds us that motion depends on the observer’s reference frame, so the same object can appear to move differently to different observers.

When solving motion problems, always pay attention to distance vs. displacement, direction, and the reference frame. These are the keys to understanding speed, velocity, and relative motion.

Acceleration and Deceleration are ideas that help us describe how motion changes. In everyday language, people often say a car is “accelerating” only when it speeds up. In science, acceleration means any change in velocity over time.

Velocity is not just speed. Velocity includes both how fast something moves and the direction it moves. This means an object can accelerate if it speeds up, slows down, or changes direction.

Deceleration usually means slowing down. In physics, slowing down is often described as negative acceleration, depending on which direction is chosen as positive.

Understanding acceleration and deceleration is important in classical mechanics because forces cause changes in motion. When a net force acts on an object, the object may speed up, slow down, or change direction.

1. What is acceleration?

Acceleration tells us how quickly velocity changes over time. The basic formula is:

$$a = \frac{\Delta v}{\Delta t}$$

Here:

• 5 is acceleration
• \(\Delta v\) means change in velocity
• \(\Delta t\) means change in time

Change in velocity is found by subtracting the starting velocity from the ending velocity:

$$\Delta v = v_f - v_i$$

So acceleration can also be written as:

$$a = \frac{v_f - v_i}{t}$$

if the time measured is the total time interval.

The standard unit for acceleration is meters per second squared, written as \(m/s^2\).

This unit may look strange at first, but it means the velocity changes by a certain number of meters per second every second. For example, an acceleration of \(2\,m/s^2\) means the velocity increases by \(2\,m/s\) each second.

2. Positive and negative acceleration

In one-dimensional motion, we usually choose one direction to be positive. For example, motion to the right might be positive, and motion to the left might be negative.

Once that choice is made, acceleration can be:

• Positive: acceleration points in the positive direction
• Negative: acceleration points in the negative direction

A negative acceleration does not always mean an object is slowing down. It depends on the direction of motion.

For example:

• If a car is moving to the right and has negative acceleration, it may slow down.
• If an object is moving to the left and also has negative acceleration, it may speed up in the left direction.

So it is better to think carefully about direction, not just the sign.

3. What is deceleration?

Deceleration means the speed of an object decreases. This happens when acceleration acts in the direction opposite to the object's motion.

For example, if a bicycle is moving forward and the rider uses the brakes, the bicycle slows down. The acceleration is opposite to the direction of motion, so this is deceleration.

In many school problems, deceleration is shown with a negative value, but that is only true if the positive direction has been chosen to match the original motion.

4. Acceleration, velocity, and time

When acceleration is constant, velocity changes in a regular way. The kinematic equation for velocity is:

$$v_f = v_i + at$$

This equation means:

• Start with the initial velocity \(v_i\)
• Add the change caused by acceleration over time, \(at\)

If acceleration is positive, velocity increases in the positive direction. If acceleration is negative, velocity changes in the negative direction.

This equation is useful when you know three of the four quantities and want to find the missing one.

5. How force relates to acceleration

Newton’s Second Law connects force and acceleration:

$$F_{net} = ma$$

This means the net force on an object causes it to accelerate. If the net force is zero, the acceleration is zero, and the object keeps moving at constant velocity or stays at rest.

If the net force acts in the same direction as motion, the object may speed up. If the net force acts opposite the motion, the object may slow down.

This is why pushing a shopping cart makes it speed up, and friction can make it slow down.

6. Reading motion in simple situations

Here are some common motion situations:

• Speeding up forward: velocity forward, acceleration forward
• Slowing down forward: velocity forward, acceleration backward
• Speeding up backward: velocity backward, acceleration backward
• Slowing down backward: velocity backward, acceleration forward

The key idea is simple: if velocity and acceleration point in the same direction, the object speeds up. If they point in opposite directions, the object slows down.

7. Worked Example 1: Finding acceleration from a change in speed

A runner increases velocity from \(2\,m/s\) to \(8\,m/s\) in \(3\,s\). Find the acceleration.

Step 1: Write the formula.

$$a = \frac{v_f - v_i}{t}$$

Step 2: Substitute the values.

$$a = \frac{8 - 2}{3}$$

Step 3: Solve.

$$a = \frac{6}{3} = 2\,m/s^2$$

Answer: The runner’s acceleration is \(2\,m/s^2\).

This means the runner’s velocity increases by \(2\,m/s\) every second.

8. Worked Example 2: Finding deceleration

A car is moving at \(20\,m/s\). It slows to \(5\,m/s\) in \(5\,s\). Find the acceleration.

Step 1: Use the formula.

$$a = \frac{v_f - v_i}{t}$$

Step 2: Substitute.

$$a = \frac{5 - 20}{5}$$

Step 3: Solve.

$$a = \frac{-15}{5} = -3\,m/s^2$$

Answer: The acceleration is \(-3\,m/s^2\).

The negative sign shows that the acceleration is opposite the chosen positive direction. Since the car is slowing down, this is also called deceleration.

9. Worked Example 3: Using the velocity equation

A skateboard starts at \(4\,m/s\) and accelerates at \(1.5\,m/s^2\) for \(6\,s\). What is its final velocity?

Step 1: Write the equation.

$$v_f = v_i + at$$

Step 2: Substitute.

$$v_f = 4 + (1.5)(6)$$

Step 3: Solve.

$$v_f = 4 + 9 = 13\,m/s$$

Answer: The final velocity is \(13\,m/s\).

Because the acceleration is positive and in the same direction as the motion, the skateboard speeds up.

10. Worked Example 4: Slowing to a stop

A ball rolls at \(10\,m/s\) and slows down with an acceleration of \(-2\,m/s^2\). How long does it take to stop?

When the ball stops, its final velocity is \(0\,m/s\).

Step 1: Use the equation.

$$v_f = v_i + at$$

Step 2: Substitute the known values.

$$0 = 10 + (-2)t$$

Step 3: Solve for \(t\).

$$0 = 10 - 2t$$

$$2t = 10$$

$$t = 5\,s$$

Answer: It takes \(5\,s\) for the ball to stop.

11. Common mistakes to avoid

• Confusing speed with velocity: velocity includes direction.
• Forgetting units: acceleration should usually be written in \(m/s^2\).
• Ignoring signs: positive and negative values matter in motion problems.
• Thinking negative acceleration always means slowing down: it depends on the direction of motion.
• Mixing up initial and final velocity: remember \(v_i\) is the starting velocity and \(v_f\) is the ending velocity.

12. Quick problem-solving steps

1. Identify what is known: \(v_i\), \(v_f\), \(a\), or \(t\).
2. Choose the correct equation.
3. Keep track of direction using positive and negative signs.
4. Substitute carefully.
5. Check whether the answer makes sense.

For example, if an object is slowing down, your answer should show a decrease in speed. If your result says the object speeds up when the problem says it slows down, check your signs again.

13. Why this matters

Acceleration and deceleration are used to understand many real-life situations. Drivers need to know how quickly a car can stop. Engineers study acceleration when designing vehicles and safety features. Athletes use acceleration to improve performance in races and sports.

These ideas also connect to larger topics in mechanics, including force, motion graphs, Newton’s laws, and energy changes in moving systems.

Summary

Acceleration is the rate at which velocity changes over time. It can happen when an object speeds up, slows down, or changes direction. Deceleration means slowing down and happens when acceleration is opposite to the motion.

The most important equations are:

$$a = \frac{v_f - v_i}{t}$$

and

$$v_f = v_i + at$$

Always pay attention to direction, signs, and units. If velocity and acceleration point in the same direction, the object speeds up. If they point in opposite directions, the object slows down.

Kinematic Graphing Analysis is the study of motion using graphs. Instead of only reading numbers from a table, we can look at how position, velocity, and time are connected visually. Graphs help us see whether an object is standing still, moving at a steady speed, speeding up, slowing down, or changing direction.

In this lesson, you will learn how to read two very important graphs:

• Position-time graphs, which tell us how location changes over time
• Velocity-time graphs, which tell us how velocity changes over time

You will also learn two key ideas:

• The slope of a position-time graph gives velocity
• The area under a velocity-time graph gives displacement, and the slope of a velocity-time graph gives acceleration

These ideas are powerful because they let you find motion quantities even when they are not written directly.

1. Position, velocity, and acceleration

Before reading graphs, let us review the three motion ideas.

• Position: where an object is located
• Velocity: how fast an object moves and in what direction
• Acceleration: how quickly velocity changes

If an object moves in a straight line, these quantities are related:

Average velocity:

Average acceleration:

Here, \(\Delta x\) means change in position, \(\Delta v\) means change in velocity, and \(\Delta t\) means change in time.

2. Reading a position-time graph

A position-time graph has time on the horizontal axis and position on the vertical axis. Each point shows where the object is at a certain time.

The most important thing to remember is this:

The slope of a position-time graph tells you the velocity.

Slope means how steep the graph is. In math, slope is:

Since \(\frac{\Delta x}{\Delta t}\) is velocity, the slope is velocity.

Here is how to understand different slopes on a position-time graph:

• Positive slope: the object has positive velocity and is moving in the positive direction
• Negative slope: the object has negative velocity and is moving in the opposite direction
• Zero slope: the object is not moving
• Steeper slope: greater speed

If the graph is a straight line, the velocity is constant. If the graph curves, the velocity is changing.

3. Reading a velocity-time graph

A velocity-time graph has time on the horizontal axis and velocity on the vertical axis. Each point shows the object’s velocity at a certain moment.

There are two very important things you can get from a velocity-time graph:

• Slope gives acceleration
• Area under the graph gives displacement

Slope on a velocity-time graph

The slope is:

Since \(\frac{\Delta v}{\Delta t}\) is acceleration, the slope gives acceleration.

• Positive slope: positive acceleration
• Negative slope: negative acceleration
• Zero slope: zero acceleration, so velocity stays constant

Area under a velocity-time graph

The area between the graph and the time axis tells the object’s displacement. Displacement means the change in position.

For simple shapes:

• Rectangle area: $$A = bh$$
• Triangle area: $$A = \frac{1}{2}bh$$

On a velocity-time graph:

• The base is the time interval
• The height is the velocity

So area becomes:

If part of the graph is below the time axis, that area counts as negative displacement because the object is moving in the negative direction.

4. Displacement vs. distance

This is an important graphing idea. Displacement includes direction. Distance does not.

• Displacement: overall change in position
• Distance: total ground covered

For example, if an object moves 5 m forward and then 2 m backward:

• Distance = \(5 + 2 = 7\) m
• Displacement = \(5 - 2 = 3\) m

On a velocity-time graph, signed area gives displacement. If you wanted total distance, you would add all the areas as positive values.

5. How to analyze graphs step by step

When you see a motion graph, use this process.

1. Check the axes. Make sure you know whether the graph is position-time or velocity-time.
2. Read units carefully. Position may be in meters, velocity in meters per second, and time in seconds.
3. Look at the shape. Straight, curved, rising, falling, or flat.
4. Use slope when needed.
   • Position-time slope \(\rightarrow\) velocity
   • Velocity-time slope \(\rightarrow\) acceleration
5. Use area when needed.
   • Velocity-time area \(\rightarrow\) displacement
6. Think about direction. Positive and negative values matter.

Worked Example 1: Velocity from a position-time graph

A runner’s position changes from 2 m at \(t=1\) s to 14 m at \(t=4\) s. Find the runner’s velocity.

Step 1: Use slope of the position-time graph.

Step 2: Find the changes.

Step 3: Calculate.

Answer: The runner’s velocity is 4 m/s.

This is a positive value, so the runner is moving in the positive direction.

Worked Example 2: Understanding a position-time graph shape

Suppose a position-time graph has three sections:

• From 0 s to 2 s, the line rises steadily
• From 2 s to 5 s, the line is flat
• From 5 s to 7 s, the line slopes downward steeply

What is happening in each section?

Section 1: Rising steadily

A straight upward line means a constant positive slope. So the object has constant positive velocity.

Section 2: Flat line

A flat line has zero slope. So the object has zero velocity and is stopped.

Section 3: Downward steep line

A downward line means negative slope, so the object has negative velocity. Because the line is steep, the speed is relatively large.

Answer:

• First, the object moves forward at constant speed
• Then, it stops
• Finally, it moves backward at a faster constant speed

Worked Example 3: Acceleration from a velocity-time graph

A car’s velocity changes from 6 m/s at \(t=2\) s to 18 m/s at \(t=5\) s. Find the acceleration.

Step 1: Use slope of the velocity-time graph.

Step 2: Find the changes.

Step 3: Calculate.

Answer: The car’s acceleration is 4 m/s².

Because the acceleration is positive, the velocity is increasing in the positive direction.

Worked Example 4: Displacement from a velocity-time graph

An object moves with a constant velocity of 5 m/s for 4 s. Find the displacement from the velocity-time graph.

On a velocity-time graph, this makes a rectangle:

• Base = 4 s
• Height = 5 m/s

Step 1: Use area of a rectangle.

Step 2: Substitute values.

Step 3: Calculate.

Answer: The displacement is 20 m.

Now suppose the graph instead shows velocity increasing from 0 m/s to 8 m/s over 4 s. This makes a triangle under the graph.

• Base = 4 s
• Height = 8 m/s

Use triangle area:

Answer: The displacement is 16 m.

6. What negative values mean

Students often think negative velocity or negative acceleration means “slowing down,” but that is not always true. Negative values describe direction on the chosen axis.

• Negative velocity means motion in the negative direction
• Negative acceleration means acceleration points in the negative direction

An object can be slowing down or speeding up with either positive or negative acceleration. You must compare the direction of velocity and acceleration.

For 9th Grade graph reading, the safest approach is:

• Use the sign to describe direction
• Use the graph shape to describe whether speed is increasing or decreasing

7. Common mistakes to avoid

• Mixing up slope and area. On a velocity-time graph, slope is acceleration and area is displacement.
• Forgetting units. Velocity should be in m/s, acceleration in m/s², and displacement in m.
• Ignoring negative values. A graph below the time axis shows motion in the negative direction.
• Using the wrong graph type. Slope on a position-time graph is velocity, not acceleration.
• Confusing displacement with distance. Signed area gives displacement.

8. Quick comparison chart

• Position-time graph
  • Slope \(\rightarrow\) velocity
  • Flat line \(\rightarrow\) object at rest
  • Straight line \(\rightarrow\) constant velocity
• Velocity-time graph
  • Slope \(\rightarrow\) acceleration
  • Area under graph \(\rightarrow\) displacement
  • Flat line \(\rightarrow\) constant velocity

9. Final summary

Kinematic graphs let us describe motion clearly. On a position-time graph, the slope tells velocity. On a velocity-time graph, the slope tells acceleration and the area under the graph tells displacement.

When solving problems, always begin by checking which variables are on the axes. Then decide whether you need a slope or an area. If you keep track of signs, units, and graph shape, you can understand motion even when no equation is given directly.

Newton's First Law and Inertial Mass

Have you ever noticed that a soccer ball starts moving easily when you kick it, but a shopping cart full of groceries is much harder to get moving? This difference helps us understand two important ideas in physics: Newton's First Law and inertial mass.

Newton's First Law explains what objects do when the forces on them are balanced. Inertial mass explains how strongly an object resists changes in motion. Together, these ideas help us understand why some objects are easy to speed up, slow down, or turn, while others are not.

Newton's First Law says:

An object will stay at rest, or keep moving at a constant velocity in a straight line, unless acted on by a net external force.

This law is sometimes called the law of inertia. The word inertia means the tendency of an object to resist changes in its motion.

There are two parts to this law:

• If an object is at rest, it will stay at rest unless a net force acts on it.
• If an object is moving, it will keep moving in the same direction at the same speed unless a net force acts on it.

This means motion does not require a constant force to keep going. Instead, a force is needed to change motion.

To understand this better, we need to know what net force means. The net force is the overall force acting on an object after all pushes and pulls are combined.

If the net force is zero, the forces are balanced. Balanced forces do not change the object's motion.

If the net force is not zero, the forces are unbalanced. Unbalanced forces cause the object to speed up, slow down, or change direction.

We can write this idea simply as:

$$F_{\text{net}} = 0 \quad \Rightarrow \quad \text{no change in motion}$$

So if an object is sitting still and the net force is zero, it stays still. If it is already moving and the net force is zero, it keeps moving with constant velocity.

Constant velocity means both the speed and direction stay the same. If either speed changes or direction changes, then the velocity changes.

What is inertial mass?

Inertial mass is a measure of how much an object resists changes in its motion. An object with more mass has more inertia, so it is harder to start moving, harder to stop, and harder to change direction.

This is why an empty wagon is easier to push than a full wagon. The full wagon has more mass, so it has more inertia.

In everyday language, we often just say mass. In this lesson, mass tells us how resistant an object is to acceleration.

Acceleration means any change in velocity. This includes:

• speeding up,
• slowing down,
• changing direction.

The connection between force, mass, and acceleration is shown by:

$$a = \frac{F_{\text{net}}}{m}$$

This equation means:

• for the same net force, a larger mass gets a smaller acceleration,
• for the same mass, a larger net force gives a larger acceleration.

So inertial mass tells us how much an object resists acceleration.

Important idea: mass does not mean an object cannot move. It means the object is harder to change from whatever motion it already has.

For example, a heavy truck moving on a road has a lot of inertia. It is harder to stop than a bicycle because its mass is much greater.

Everyday examples of Newton's First Law

• A book on a desk stays still until someone pushes it.
• A hockey puck slides across smooth ice for a long time because there is very little friction.
• When a car stops suddenly, passengers move forward because their bodies keep their previous motion.
• Seat belts help by providing the force needed to stop the passengers safely.

Friction is important in real life. If you slide a box across the floor, it eventually stops. That does not mean Newton's First Law is wrong. It means a force, usually friction, acts on the box and changes its motion.

Without friction, the box would keep moving at constant velocity.

Balanced and unbalanced forces

Suppose two students push a box from opposite sides with equal force. If one pushes with 20 N to the right and the other pushes with 20 N to the left, the net force is:

$$F_{\text{net}} = 20\,\text{N} - 20\,\text{N} = 0\,\text{N}$$

The forces are balanced. The box will not change its motion.

If one student pushes with 30 N to the right and the other pushes with 10 N to the left, then:

$$F_{\text{net}} = 30\,\text{N} - 10\,\text{N} = 20\,\text{N to the right}$$

The forces are unbalanced, so the box accelerates to the right.

Worked Example 1: Identifying Newton's First Law

A soccer ball is resting on the field. No one touches it. What happens?

Step 1: Think about the forces.

The ball has gravity pulling down and the ground pushing up. These forces balance.

Step 2: Find the net force.

The net force is zero.

Step 3: Apply Newton's First Law.

Because the net force is zero, the ball stays at rest.

Answer: The soccer ball remains still unless an unbalanced force, like a kick, acts on it.

Worked Example 2: Comparing inertia

A student pushes two carts with the same force. Cart A has a mass of 10 kg. Cart B has a mass of 30 kg. Which cart changes motion more easily?

Step 1: Compare the masses.

Cart A has less mass than Cart B.

Step 2: Use the idea of inertia.

Less mass means less inertia. More mass means more inertia.

Step 3: Decide which cart resists acceleration less.

Cart A resists the change in motion less than Cart B.

Answer: Cart A changes motion more easily because it has less inertial mass.

Worked Example 3: Calculating acceleration from force and mass

A net force of 12 N acts on a 3 kg object. Find the acceleration.

Step 1: Write the formula.

$$a = \frac{F_{\text{net}}}{m}$$

Step 2: Substitute the values.

$$a = \frac{12\,\text{N}}{3\,\text{kg}}$$

Step 3: Calculate.

$$a = 4\,\text{m/s}^2$$

Answer: The object accelerates at \(4\,\text{m/s}^2\).

This example shows that when mass is not very large, a given force can produce a noticeable acceleration.

Worked Example 4: Same force, different masses

A net force of 20 N is applied to two objects.

• Object 1 has mass \(5\,\text{kg}\).
• Object 2 has mass \(10\,\text{kg}\).

Find the acceleration of each object and decide which has more inertia.

Step 1: Use the formula for Object 1.

$$a = \frac{F_{\text{net}}}{m} = \frac{20}{5} = 4\,\text{m/s}^2$$

Step 2: Use the formula for Object 2.

$$a = \frac{F_{\text{net}}}{m} = \frac{20}{10} = 2\,\text{m/s}^2$$

Step 3: Compare the results.

Object 1 gets a larger acceleration. Object 2 gets a smaller acceleration.

Step 4: Connect to inertia.

Object 2 has more mass, so it has more inertia.

Answer: Object 1 accelerates at \(4\,\text{m/s}^2\), Object 2 accelerates at \(2\,\text{m/s}^2\), and Object 2 has more inertial mass.

Common misunderstandings

• Misunderstanding 1: Moving objects need a force to keep moving.
  Actually, moving objects only need a force if their motion is changing. If the net force is zero, they keep moving at constant velocity.
• Misunderstanding 2: Heavier objects always move slower.
  Mass does not automatically decide speed. Mass tells how hard it is to change the motion.
• Misunderstanding 3: If an object stops, there was no inertia.
  All objects with mass have inertia. An object stops because a force such as friction acts on it.

How seat belts connect to inertia

When a car is moving, the passengers are moving with it. If the car stops suddenly, the passengers' bodies tend to keep moving forward because of inertia.

The seat belt provides the unbalanced force that brings the passenger to rest with the car. This is a real-life example of Newton's First Law.

Key points to remember

• Newton's First Law describes what happens when the net force is zero.
• An object at rest stays at rest unless acted on by a net external force.
• An object in motion stays in motion with constant velocity unless acted on by a net external force.
• Inertia is the tendency to resist changes in motion.
• Mass is a measure of inertia.
• More mass means more resistance to acceleration.
• The relationship between force, mass, and acceleration is $$a = \frac{F_{\text{net}}}{m}$$

Brief Summary

Newton's First Law tells us that objects do not change their motion unless a net force acts on them. If forces are balanced, the object stays at rest or keeps moving at constant velocity.

Inertial mass measures how much an object resists changes in motion. Objects with greater mass have greater inertia, so they are harder to speed up, slow down, or turn. Understanding these ideas helps explain many everyday events, from pushing carts to wearing seat belts in cars.

Newton's Second Law: Force, Mass, and Acceleration

When an object speeds up, slows down, or changes direction, it is accelerating. Newton's Second Law explains what causes that acceleration. It connects three important ideas in motion: force, mass, and acceleration.

This law helps us answer questions like: Why does an empty shopping cart speed up more easily than a full one? Why does a stronger push make something move faster? Newton's Second Law gives a simple rule for all of these situations.

The law is written as:

$$F = ma$$

In this equation:

• (F) is the net force on an object, measured in newtons (N)
• (m) is the mass of the object, measured in kilograms (kg)
• (a) is the acceleration of the object, measured in meters per second squared, or (m/s^2)

This means that the acceleration of an object depends on two things:

• How much net force is acting on it
• How much mass it has

A larger net force causes a larger acceleration. A larger mass causes a smaller acceleration, if the force stays the same.

Net force means the overall force after adding all forces acting on the object. Forces in the same direction add together. Forces in opposite directions subtract.

So a more complete way to think about the law is:

$$F_{net} = ma$$

If the net force is zero, then the acceleration is zero. That means the object will either stay at rest or keep moving at a constant speed in a straight line.

How force, mass, and acceleration are related

Newton's Second Law shows two key patterns.

• If mass stays the same, increasing the net force increases the acceleration.
• If net force stays the same, increasing the mass decreases the acceleration.

For example, if you push two objects with the same force, the object with less mass will accelerate more. If you push the same object harder, it will accelerate more.

This is why it is easier to start moving a bicycle than a car. The car has much more mass, so it needs much more force to get the same acceleration.

Units in Newton's Second Law

It is important to use the correct units:

• Force: newtons, (N)
• Mass: kilograms, (kg)
• Acceleration: meters per second squared, (m/s^2)

One newton is the amount of force needed to accelerate a mass of 1 kilogram at (1 \, m/s^2) .

So:

$$1\,N = 1\,kg \cdot m/s^2$$

Rearranging the formula

You can use the same equation to solve for different unknowns.

• To find force: (F = ma)
• To find acceleration: (a = \frac{F}{m})
• To find mass: (m = \frac{F}{a})

Always make sure you are using the net force, not just one force from the problem.

Finding net force

Many problems have more than one force acting on an object. Before using Newton's Second Law, find the net force.

Suppose one student pushes a box to the right with (20\,N) , and another student pushes it to the left with (5\,N) . The forces are in opposite directions, so subtract:

$$F_{net} = 20\,N - 5\,N = 15\,N \text{ to the right}$$

That net force is what causes the acceleration.

Worked Example 1: Finding force

A (4\,kg) cart accelerates at (3\,m/s^2) . What net force acts on the cart?

Step 1: Write the formula

$$F = ma$$

Step 2: Substitute the values

$$F = (4\,kg)(3\,m/s^2)$$

Step 3: Multiply

$$F = 12\,N$$

Answer: The net force is 12 N.

Worked Example 2: Finding acceleration

A (10\,kg) object is pulled with a net force of (25\,N) . What is its acceleration?

Step 1: Use the formula for acceleration

$$a = \frac{F}{m}$$

Step 2: Substitute the values

$$a = \frac{25\,N}{10\,kg}$$

Step 3: Divide

$$a = 2.5\,m/s^2$$

Answer: The acceleration is 2.5 m/s^2.

Worked Example 3: Finding net force first, then acceleration

A (6\,kg) sled is pulled to the right with (18\,N) . Friction pushes back to the left with (6\,N) . What is the sled's acceleration?

Step 1: Find the net force

The forces are in opposite directions, so subtract:

$$F_{net} = 18\,N - 6\,N = 12\,N$$

Step 2: Use Newton's Second Law

$$a = \frac{F_{net}}{m}$$

Step 3: Substitute the values

$$a = \frac{12\,N}{6\,kg}$$

Step 4: Divide

$$a = 2\,m/s^2$$

Answer: The sled accelerates at 2 m/s^2 to the right.

Worked Example 4: Finding mass

A machine applies a net force of (48\,N) to an object. The object accelerates at (6\,m/s^2) . What is the object's mass?

Step 1: Rearrange the formula

$$m = \frac{F}{a}$$

Step 2: Substitute the values

$$m = \frac{48\,N}{6\,m/s^2}$$

Step 3: Divide

$$m = 8\,kg$$

Answer: The mass is 8 kg.

Common mistakes to avoid

• Using total force instead of net force: If forces act in opposite directions, subtract them first.
• Mixing up mass and weight: Mass is measured in kilograms. Weight is a force and is measured in newtons.
• Forgetting units: Always include (N) , (kg) , or (m/s^2) in your answer.
• Rearranging the formula incorrectly: Check whether you need to multiply or divide.

Real-life examples

• An empty grocery cart accelerates more than a full cart when pushed with the same force.
• A soccer ball accelerates quickly when kicked because it has a small mass.
• A truck needs much more force than a skateboard to get the same acceleration.

How to solve Newton's Second Law problems

1. Read the problem carefully.
2. Identify what is given: force, mass, acceleration, or more than one force.
3. Find the net force if needed.
4. Choose the correct form of the formula.
5. Substitute the values with units.
6. Solve and write the final answer with units.

Brief Summary

Newton's Second Law states that the net force on an object equals its mass times its acceleration: (F_{net} = ma) . More net force causes more acceleration, while more mass causes less acceleration if the force stays the same. To solve problems, first find the net force, then use the formula to calculate force, mass, or acceleration.

Newton's Second Law helps us understand how things move when we push or pull them.

A force is a push or a pull. When you kick a ball, open a door, or pull a wagon, you are using force.

Newton's Second Law tells us something very important: a bigger push or pull makes something speed up more, and a heavier thing is harder to speed up.

Scientists write this idea like this:

$$F = m \times a$$

Here is what the letters mean:

• F = force, or push/pull
• m = mass, or how much stuff is in an object
• a = acceleration, or how much the speed changes

For 2nd grade, you can think of it this way:

• More force = more speeding up
• More mass = less speeding up

If two toys are the same size, but one is much heavier, the heavier one needs a stronger push to move faster.

If two objects weigh the same, the one you push harder will speed up more.

Let's break the idea into simple parts.

1. A stronger push makes a bigger change in motion.

Imagine two soccer balls. You tap one softly. You kick the other one hard. The ball you kick hard rolls away faster. That is Newton's Second Law.

2. Heavy things are harder to move fast.

Think about pushing an empty wagon and a wagon full of books. The full wagon has more mass. It does not speed up as easily. You need more force to make it go faster.

3. Force and mass work together.

A small push may move a light object a lot. The same small push may barely move a heavy object. That is because mass matters.

What does acceleration mean?

Acceleration means a change in speed. If something starts moving faster, slows down, or changes how it moves because of a push or pull, that is acceleration.

For this lesson, we will mostly think of acceleration as speeding up.

Real-life examples

• Pushing a toy car harder makes it go faster.
• Pushing a big box is harder than pushing a small box.
• Kicking a rubber ball makes it speed away.
• A grocery cart full of food needs more push than an empty cart.

Worked Example 1: Same object, different pushes

Ben has one toy car. First, he gives it a small push. Then, he gives it a big push.

What happens?

The big push gives more force, so the toy car speeds up more.

Answer: The toy car goes faster with the big push.

Worked Example 2: Same push, different masses

Lia pushes a small ball and a heavy bowling ball with the same amount of force.

What happens?

The small ball has less mass, so it speeds up more. The bowling ball has more mass, so it speeds up less.

Answer: The small ball moves faster because it is lighter.

Worked Example 3: Empty wagon and full wagon

Jay pushes an empty wagon. Then he pushes a wagon filled with sand using the same push.

What happens?

The empty wagon has less mass, so it speeds up more easily. The full wagon has more mass, so it needs more force to speed up the same way.

Answer: The empty wagon moves faster with the same push.

Worked Example 4: Using the math idea

Scientists use this rule:

$$F = m \times a$$

This means force equals mass times acceleration.

Let's use easy numbers. If mass is 2 and acceleration is 3, then:

$$F = 2 \times 3$$

$$F = 6$$

Answer: The force is 6.

You do not need to worry about big number names. The important idea is this: if mass gets bigger or acceleration gets bigger, force must also get bigger.

Easy ways to remember Newton's Second Law

• Push harder, go faster.
• Heavier things need bigger pushes.
• Light things move more easily.

Let's compare.

1. A little push on a light toy = big change in motion
2. A little push on a heavy box = small change in motion
3. A big push on a heavy box = bigger change in motion

Things to watch out for

• Do not think all objects move the same with the same push.
• Do not forget that heavier objects need more force.
• Do not forget that stronger forces make bigger changes in speed.

Try thinking about these questions.

• If you push your scooter harder, what happens?
• If you push a backpack and a pencil box with the same force, which speeds up more?
• Why is a full shopping cart harder to get moving than an empty one?

Summary

Newton's Second Law explains how force, mass, and acceleration are connected.

A stronger force makes an object speed up more. A greater mass makes an object harder to speed up.

We can write the law like this: $$F = m \times a$$

Remember: more push means more speed-up, and more mass means you need more push.

Kinematic Equations are math rules we use to describe how objects move in a straight line when their acceleration stays the same.

These equations help us answer questions like:

• How fast is something moving after a certain time?
• How far has it traveled?
• How long did it take to reach a certain speed?

In this lesson, you will learn what each part of the equations means, when to use them, and how to solve motion problems step by step.

First, let’s review some important motion words.

• Position: where an object is
• Displacement: how far an object moves from its starting point, including direction
• Velocity: speed in a direction
• Acceleration: how quickly velocity changes
• Time: how long the motion lasts

For kinematic equations, we usually use these symbols:

• Initial velocity: \(v_i\) = starting velocity
• Final velocity: \(v_f\) = ending velocity
• Acceleration: \(a\)
• Time: \(t\)
• Displacement: \(d\)

The kinematic equations only work when acceleration is constant.

That means the acceleration does not keep changing. For example, if a toy car speeds up by the same amount each second, then constant acceleration is happening.

If acceleration changes in an uneven way, these equations do not work as well.

Here are the main kinematic equations.

1. To find final velocity:

This equation means the final velocity equals the starting velocity plus the change caused by acceleration over time.

2. To find displacement when time is known:

This equation tells how far an object moves while starting with some velocity and accelerating.

3. To connect velocity and displacement without time:

This equation is useful when time is missing from the problem.

How do you know which equation to use?

A good strategy is to list what you know and what you need to find.

• If you know \(v_i\), \(a\), and \(t\), and need \(v_f\), use \(v_f = v_i + at\).
• If you know \(v_i\), \(a\), and \(t\), and need \(d\), use \(d = v_i t + \frac{1}{2}at^2\).
• If you know \(v_i\), \(a\), and \(d\), and need \(v_f\), use \(v_f^2 = v_i^2 + 2ad\).

Direction matters.

In motion problems, positive and negative signs are important. If we choose forward as positive, then backward is negative.

If an object is slowing down while moving forward, its acceleration is negative.

For example:

• Moving to the right: positive velocity
• Moving to the left: negative velocity
• Speeding up to the right: positive acceleration
• Slowing down while moving to the right: negative acceleration

Worked Example 1: Finding final velocity

A scooter starts at \(2\text{ m/s}\) and accelerates at \(3\text{ m/s}^2\) for \(4\text{ s}\). What is its final velocity?

Step 1: Write what you know.

• \(v_i = 2\text{ m/s}\)
• \(a = 3\text{ m/s}^2\)
• \(t = 4\text{ s}\)

Step 2: Choose the equation.

Step 3: Substitute the values.

Answer: The final velocity is \(14\text{ m/s}\).

Worked Example 2: Finding displacement

A ball rolls with an initial velocity of \(1\text{ m/s}\) and accelerates at \(2\text{ m/s}^2\) for \(5\text{ s}\). How far does it travel?

Step 1: Write what you know.

• \(v_i = 1\text{ m/s}\)
• \(a = 2\text{ m/s}^2\)
• \(t = 5\text{ s}\)

Step 2: Choose the equation.

Step 3: Substitute the values.

Answer: The ball travels \(30\text{ m}\).

Worked Example 3: Finding final velocity without time

A bicycle starts at \(3\text{ m/s}\) and accelerates at \(4\text{ m/s}^2\) over a displacement of \(8\text{ m}\). What is its final velocity?

Step 1: Write what you know.

• \(v_i = 3\text{ m/s}\)
• \(a = 4\text{ m/s}^2\)
• \(d = 8\text{ m}\)

Step 2: Choose the equation.

Step 3: Substitute the values.

Step 4: Take the square root.

Answer: The final velocity is about \(8.5\text{ m/s}\).

Worked Example 4: Slowing down

A runner is moving at \(10\text{ m/s}\) and slows down at \(-2\text{ m/s}^2\) for \(3\text{ s}\). What is the runner’s final velocity?

Step 1: Write what you know.

• \(v_i = 10\text{ m/s}\)
• \(a = -2\text{ m/s}^2\)
• \(t = 3\text{ s}\)

Step 2: Use the equation.

Step 3: Substitute.

Answer: The runner’s final velocity is \(4\text{ m/s}\).

Common mistakes to avoid

• Mixing up velocity and acceleration: velocity tells how fast and in what direction; acceleration tells how velocity changes.
• Forgetting units: write units like meters \((\text{m})\), seconds \((\text{s})\), meters per second \((\text{m/s})\), and meters per second squared \((\text{m/s}^2)\).
• Ignoring negative signs: a negative sign can show opposite direction or slowing down.
• Using the wrong equation: always check what information is given and what is missing.

Helpful problem-solving steps

1. Read the problem carefully.
2. List the known values.
3. Decide what you need to find.
4. Pick the equation that matches.
5. Substitute the numbers carefully.
6. Solve and check if the answer makes sense.

Why kinematic equations matter

These equations are useful in real life. They can describe the motion of cars, bicycles, falling objects, balls, and even amusement park rides.

Scientists and engineers use them to predict motion and design safe machines and structures.

Brief Summary

Kinematic equations describe straight-line motion when acceleration is constant. The main quantities are initial velocity, final velocity, acceleration, time, and displacement.

The three main equations are:

When solving problems, identify what you know, choose the correct equation, and pay close attention to direction and units.

Newton's Third Law: Action-Reaction Pairs

When two objects interact, they always push or pull on each other. This idea is described by Newton's Third Law of Motion.

Newton's Third Law says: For every action force, there is an equal and opposite reaction force.

In symbols, this can be written as:

$$F_{A\to B} = -F_{B\to A}$$

This means that if object A pushes on object B with a certain force, then object B pushes back on object A with the same size force, but in the opposite direction.

This law is often confusing because students sometimes think the two forces cancel each other out. They do not cancel each other if they act on different objects. That is the key idea of this lesson.

1. What is an action-reaction pair?

An action-reaction pair is a pair of forces that two interacting objects exert on each other.

• The two forces are always the same size.
• The two forces are always in opposite directions.
• The two forces act on different objects.
• The two forces happen at the same time.

If one object is pulling, the other object is also pulling back. If one object is pushing, the other object is also pushing back.

2. Action-reaction pairs vs. balanced forces

This is the most important difference to understand.

Balanced forces are forces on the same object that are equal in size and opposite in direction. Balanced forces can cancel and cause no change in motion.

Action-reaction forces are forces on different objects. Because they act on different objects, they do not cancel each other.

For example, a book resting on a table has two forces on the book:

• The Earth's gravitational force pulls the book downward.
• The table's support force pushes the book upward.

These two forces can be balanced if the book is not moving. But they are not an action-reaction pair, because both forces act on the book.

The reaction pair for the table pushing up on the book is the book pushing down on the table.

The reaction pair for Earth pulling down on the book is the book pulling up on Earth.

3. How to identify an action-reaction pair

Use these questions:

1. Are there two interacting objects?
2. Is one force exerted by the first object on the second?
3. Is the other force exerted by the second object on the first?
4. Are the forces equal in size and opposite in direction?
5. Do the forces act on different objects?

If the answer to all of these is yes, then you have an action-reaction pair.

4. Why equal forces do not always cause equal motion

If the forces are equal, why does one object sometimes move more than the other?

The answer is that the two forces act on different masses. A small object may change motion a lot, while a very large object changes motion only a little.

For example, when you push on a wall, the wall pushes back on you with an equal force. But the wall usually does not move because it is attached strongly to the building and the ground. Your body may move backward or feel pressure in your hands instead.

5. Common examples of Newton's Third Law

• Walking: Your foot pushes backward on the ground, and the ground pushes forward on your foot.
• Swimming: You push water backward, and the water pushes you forward.
• Jumping: Your legs push down on the ground, and the ground pushes you up.
• Rocket motion: The rocket pushes gas downward, and the gas pushes the rocket upward.
• Hitting a ball: The bat pushes the ball, and the ball pushes the bat.

6. Worked Example 1: A student pushes on a wall

Situation: A student pushes on a wall with a force of \(50\,\text{N}\).

Question: What force does the wall exert on the student?

Step 1: Identify the interaction.

The student pushes on the wall.

Step 2: Apply Newton's Third Law.

The wall must push back on the student with the same size force in the opposite direction.

Answer: The wall exerts a force of \(50\,\text{N}\) on the student in the opposite direction.

So if the student pushes the wall to the right, the wall pushes the student to the left with \(50\,\text{N}\).

7. Worked Example 2: A book resting on a table

Situation: A book sits still on a table.

Question: Which forces are balanced, and which forces form action-reaction pairs?

Step 1: Look at the forces on the book.

• Gravity pulls the book downward.
• The table pushes the book upward.

Because the book is not moving up or down, these forces on the book are balanced.

Step 2: Identify the action-reaction pairs.

• Table pushes up on book ↔ book pushes down on table
• Earth pulls down on book ↔ book pulls up on Earth

Answer: The upward table force and downward gravity force are balanced forces on the same object. They are not a third-law pair. The third-law pairs involve the two interacting objects pushing or pulling on each other.

8. Worked Example 3: Walking forward

Situation: A person walks forward across the floor.

Question: What is the action-reaction pair that helps the person move?

Step 1: Think about what the foot does.

The foot pushes backward on the ground.

Step 2: Apply Newton's Third Law.

The ground pushes forward on the foot with an equal force.

Answer: The action-reaction pair is:

• Foot pushes backward on ground
• Ground pushes forward on foot

This forward push from the ground helps the person move forward.

9. Worked Example 4: A bat hits a baseball

Situation: A bat hits a baseball with a force of \(120\,\text{N}\).

Question: What force does the baseball exert on the bat?

Step 1: Identify the force from the bat to the ball.

The bat exerts \(120\,\text{N}\) on the baseball.

Step 2: Use Newton's Third Law.

The baseball exerts an equal and opposite force on the bat.

Answer: The baseball exerts \(120\,\text{N}\) on the bat in the opposite direction.

This is why the batter feels the impact in their hands.

10. Common mistakes to avoid

• Mistake 1: Thinking action-reaction forces cancel each other. They do not, because they act on different objects.
• Mistake 2: Matching two forces on the same object as a third-law pair. Those may be balanced forces instead.
• Mistake 3: Thinking the bigger object exerts a bigger force. In an interaction, both objects exert equal-sized forces on each other.
• Mistake 4: Thinking one force happens first and the other happens later. Action and reaction happen at the same time.

11. Quick check questions

Try these on your own:

1. If you kick a soccer ball, what force pair is involved?
2. If a chair supports a sitting student, what force on the student balances gravity?
3. Is that support force and gravity a third-law pair? Why or why not?
4. When a rocket rises, what does it push, and what pushes it back?

12. Brief summary

Newton's Third Law explains forces between interacting objects. If object A pushes or pulls on object B, then object B pushes or pulls on object A with an equal force in the opposite direction.

Remember the key rule: action-reaction pairs act on different objects. Balanced forces act on the same object. If you keep that difference clear, it becomes much easier to identify force pairs correctly.

Free-Body Diagrams and Vector Resolution help us understand how forces act on an object and how those forces combine to change its motion.

In this lesson, you will learn how to identify forces, draw a free-body diagram, break angled forces into horizontal and vertical parts, and find the net force on an object.

This is an important skill in mechanics because objects are often pulled, pushed, or supported in more than one direction at the same time. A clear diagram makes the physics much easier to understand.

1. What is a force?

A force is a push or a pull. Forces are measured in newtons, written as N.

Forces are also vectors. That means they have:

• magnitude — how strong the force is
• direction — which way the force acts

For example, a force of 10 N to the right is different from a force of 10 N upward.

2. What is a free-body diagram?

A free-body diagram is a simple drawing that shows all the forces acting on one object.

Usually, the object is shown as a dot or a small box. Then arrows are drawn from the object to show each force. The length of each arrow can be scaled to represent the size of the force.

A good free-body diagram includes:

• only one object
• all forces acting on that object
• correct force directions
• labels for each force

3. Common forces in 9th Grade mechanics

Here are the most common forces you will see.

• Weight \, or gravitational force: pulls an object downward toward Earth. It is often written as \(F_g\) or \(W\).
• Normal force: the support force from a surface. It acts perpendicular to the surface.
• Tension force: the pulling force from a rope, string, or cable.
• Applied force: a push or pull from a person or another object.
• Friction force: a force that opposes motion or attempted motion between surfaces in contact.

4. Understanding each force

Weight always acts downward. Near Earth, weight can be found using:

where:

• \(W\) is weight in newtons
• \(m\) is mass in kilograms
• \(g\) is gravitational field strength, about \(9.8\,\text{m/s}^2\), often rounded to \(10\,\text{m/s}^2\) in school problems

Normal force is not always equal to weight. It depends on how the surface pushes back on the object. On a flat surface with no vertical acceleration, the normal force often equals the weight.

Friction acts parallel to the surface and opposes motion. If a box is sliding to the right, friction acts to the left.

Tension acts along a rope or string, pulling away from the object.

Applied force can act in any direction depending on how the object is pushed or pulled.

5. Steps for drawing a free-body diagram

1. Choose one object to study.
2. Draw the object as a box or dot.
3. Identify every force acting on it.
4. Draw arrows showing the direction of each force.
5. Label each arrow clearly.
6. Do not draw forces the object exerts on other objects. Only include forces acting on the chosen object.

6. Balanced and unbalanced forces

If forces cancel out, the net force is zero. These are balanced forces.

If the forces do not cancel, there is a net force. These are unbalanced forces.

The net force tells us the overall effect of all the forces together.

• If net force is zero, the object stays at rest or keeps moving at constant speed in a straight line.
• If net force is not zero, the object's motion changes.

7. What is vector resolution?

Sometimes a force acts at an angle. This can make problems harder to solve.

Vector resolution means breaking one angled force into two simpler parts:

• a horizontal component
• a vertical component

These components are usually at right angles to each other.

Instead of working with one diagonal force, we work with two straight forces. This makes it easier to compare them with other horizontal and vertical forces.

8. Resolving a vector into components

Suppose a force \(F\) acts at an angle \(\theta\) above the horizontal.

Its components are:

where:

• \(F_x\) is the horizontal part
• \(F_y\) is the vertical part

If the angle is measured from the horizontal, cosine gives the horizontal side and sine gives the vertical side.

A useful memory aid is:

• cosine for the side next to the angle
• sine for the side opposite the angle

9. Sign and direction matter

When working with components, it helps to choose positive directions. A common choice is:

• right is positive \((+)\)
• up is positive \((+)\)
• left is negative \((- )\)
• down is negative \((- )\)

Then you can add forces carefully. For example, if an object has 12 N to the right and 5 N to the left, the net horizontal force is:

So the net force is 7 N to the right.

10. Scaled vector models

A scaled vector model is a force diagram where arrow lengths match the sizes of the forces using a scale.

For example, you might choose:

• 1 cm = 5 N

Then:

• a 10 N force is drawn as 2 cm
• a 20 N force is drawn as 4 cm

Scaled diagrams help you compare forces visually and check whether your answers make sense.

11. Worked Example 1: Box on a flat floor

A 5 kg box rests on a flat floor. A person pushes it with 12 N to the right. Friction is 4 N to the left. Draw the free-body diagram and find the net force.

Step 1: Identify the forces

• Weight downward
• Normal force upward
• Applied force 12 N to the right
• Friction 4 N to the left

Step 2: Find the weight

So the weight is 50 N downward.

Because the box is on a flat floor and not moving up or down, the normal force is 50 N upward.

Step 3: Horizontal net force

Step 4: Vertical net force

Answer: The net force is 8 N to the right.

In the free-body diagram, the upward and downward arrows would be equal in size, and the rightward arrow would be longer than the leftward one.

12. Worked Example 2: Pulling with a rope at an angle

A rope pulls a sled with a force of 20 N at an angle of \(30^\circ\) above the horizontal. Find the horizontal and vertical components of the pull.

Step 1: Use the component formulas

Step 2: Substitute the values

Using calculator values:

• \(\cos30^\circ \approx 0.87\)
• \(\sin30^\circ = 0.5\)

Answer:

• Horizontal component: 17.4 N to the right
• Vertical component: 10 N upward

This means the angled pull does two things at once: it pulls the sled forward and also lifts up slightly.

13. Worked Example 3: Net force with an angled pull and friction

A box is pulled across a floor by a 30 N force at \(40^\circ\) above the horizontal. Friction is 12 N to the left. Find the net horizontal force.

Step 1: Resolve the applied force horizontally

Using \(\cos40^\circ \approx 0.77\):

So the forward part of the pull is 23.1 N to the right.

Step 2: Include friction

Answer: The net horizontal force is 11.1 N to the right.

The vertical part of the pulling force would affect the normal force, but if the question asks only for horizontal net force, we focus on the horizontal forces.

14. Worked Example 4: Tension holding an object still

A 2 kg object hangs motionless from a vertical rope. Draw the free-body diagram and find the tension.

Step 1: Identify the forces

• Weight downward
• Tension upward

Step 2: Find the weight

Step 3: Use the fact that the object is motionless

If it is motionless, the net force is zero. So the upward tension must equal the downward weight.

Answer: The tension in the rope is 20 N upward.

15. Inclined surfaces: a simple idea

Sometimes an object is on a slope. Then the normal force is still perpendicular to the surface, not straight up.

The weight still acts straight down. Because the directions are different, slope problems often need vector resolution too.

At this level, the key idea is to draw forces in their correct directions:

• weight straight down
• normal force perpendicular to the slope
• friction along the slope, opposite motion
• applied force or tension in the direction of the push or pull

16. Common mistakes to avoid

• Forgetting weight — almost every object near Earth has weight.
• Drawing extra forces — only include forces acting on the chosen object.
• Confusing mass and weight — mass is in kg, weight is in N.
• Putting friction in the wrong direction — friction opposes motion or attempted motion.
• Mixing up sine and cosine — check whether the angle is measured from the horizontal or vertical.
• Assuming normal force always equals weight — this is only true in some situations.

17. Problem-solving strategy

When solving force problems, use this method:

1. Choose the object.
2. Draw the free-body diagram.
3. Label all known forces.
4. If a force is angled, resolve it into components.
5. Add forces in the horizontal direction.
6. Add forces in the vertical direction.
7. State the net force with both size and direction.

18. Quick check questions

• If a box is pushed right and friction acts left, which force opposes the motion? Friction.
• If an object is sitting still on a flat table, what two vertical forces usually act on it? Weight downward and normal force upward.
• If a force acts at an angle, why do we resolve it? To find its horizontal and vertical parts.
• What does a free-body diagram show? All the forces acting on one object.

19. Summary

A free-body diagram is a simple drawing that shows all the forces acting on one object. It helps you see which forces balance and which do not.

Forces are vectors, so they have both size and direction. When a force acts at an angle, you can resolve it into horizontal and vertical components using sine and cosine.

Once you have the components, add forces in each direction to find the net force. This is the key to understanding how pushes, pulls, tension, friction, weight, and normal force affect motion.

Frictional Forces: Static, Kinetic, and Drag

When objects move, they usually do not keep moving forever. One major reason is friction. Friction is a force that opposes motion, or opposes the attempt to move.

Friction is extremely important in everyday life. It helps your shoes grip the ground, allows car tires to move safely on roads, and lets you hold a pencil without it slipping. At the same time, friction can also make motion harder by slowing objects down and turning some of their energy into heat.

In this lesson, you will learn about three main types of frictional forces:

• Static friction — friction that prevents motion from starting
• Kinetic friction — friction acting when surfaces slide past each other
• Drag — friction from moving through a fluid such as air or water

You will also learn how friction is connected to tiny bumps on surfaces, called asperities, and how friction changes energy in a system.

1. What causes friction?

No surface is perfectly smooth, even if it looks smooth. If you zoomed in very closely, you would see tiny rough spots and bumps. These small irregularities are called asperities.

When two surfaces touch, their asperities press against each other. This makes it harder for the surfaces to move. The rough spots can catch, press, and resist sliding. That resistance is part of what creates friction.

Friction also depends on how hard the surfaces are pushed together. A heavier object usually has more friction because the surfaces are pressed together more strongly.

As friction acts, some of the object’s motion energy is transformed into thermal energy, which means heat. That is why your hands get warm if you rub them together.

2. Static friction

Static friction is the friction force that keeps an object at rest when you try to move it. It acts before the object starts sliding.

Imagine pushing a heavy box on the floor. At first, the box does not move. That does not mean there is no force. It means static friction is matching your push.

If you push gently, static friction is small. If you push harder, static friction increases. It keeps increasing until it reaches a maximum static friction. If your push becomes greater than this maximum, the box starts moving.

So static friction is special because it can change its size depending on the applied force, up to a limit.

The largest possible static friction is often written as:

$$f_s^{\text{max}} = \mu_s N$$

In this formula:

• \(f_s^{\text{max}}\) = maximum static friction
• \(\mu_s\) = coefficient of static friction
• \(N\) = normal force, the support force pushing up from the surface

The coefficient of friction is a number that describes how much grip two surfaces have. A larger value means more friction.

3. Kinetic friction

Once an object is sliding, the friction usually changes to kinetic friction. This is the friction force between surfaces that are moving past each other.

Kinetic friction usually has a nearly constant value and is often a little smaller than maximum static friction. That is why getting a box moving can be harder than keeping it moving.

The kinetic friction force is written as:

$$f_k = \mu_k N$$

In this formula:

• \(f_k\) = kinetic friction
• \(\mu_k\) = coefficient of kinetic friction
• \(N\) = normal force

Because \(\mu_k\) is often less than \(\mu_s\), sliding friction is usually less than the maximum force needed to start motion.

4. The normal force and why it matters

The normal force is the support force from a surface. On a flat floor, if an object is not accelerating up or down, the normal force is usually equal to the object’s weight.

Weight is found by:

$$W = mg$$

where:

• \(m\) = mass in kilograms
• \(g\) = gravitational field strength, about \(9.8\,\text{m/s}^2\) on Earth

On a level surface, this often means:

$$N = mg$$

If the normal force gets larger, friction usually gets larger too. This is why a full shopping cart is harder to push than an empty one.

5. Drag: friction in fluids

Drag is a type of friction that acts on objects moving through a fluid. A fluid can be a liquid, like water, or a gas, like air.

Air resistance is a common example of drag. When a cyclist moves forward, air pushes back against the motion. A swimmer also feels drag from water.

Drag depends on several things:

• the object’s speed
• the shape of the object
• the size of the object
• the type of fluid it moves through

In general, drag gets larger when speed increases. This is why it is much harder to ride a bike very fast than slowly.

Smooth, streamlined shapes reduce drag. That is why airplanes, fish, and racing cars are designed with shapes that let fluid flow around them more easily.

6. Direction of frictional forces

Friction always acts in a direction that opposes motion or opposes the tendency to move.

• If you push a box to the right and it is not moving, static friction acts to the left.
• If the box slides to the right, kinetic friction acts to the left.
• If a ball moves forward through the air, drag acts backward.

This idea is very important when drawing force diagrams or deciding whether an object will speed up, slow down, or stay still.

7. Friction and Newton’s laws

Newton’s laws help explain what friction does to motion.

Newton’s First Law says an object will stay at rest or keep moving at constant velocity unless acted on by an unbalanced force. Friction is often the unbalanced force that causes moving objects to slow down.

Newton’s Second Law says:

$$F_{\text{net}} = ma$$

If friction is the only horizontal force, then the net force points opposite the motion, so the object slows down.

If you push an object forward, you must compare your push to the friction force:

• If your push is less than maximum static friction, the object stays still.
• If your push is greater than maximum static friction, the object starts moving.
• Once moving, the acceleration depends on the difference between the applied force and kinetic friction.

8. Friction and energy transformations

Friction does not destroy energy. Instead, it changes energy from one form into another.

For example, when a sliding book slows down, its kinetic energy decreases. That energy is mostly transformed into thermal energy in the book and the floor.

This is why friction is often described as a force that dissipates useful motion energy as heat. In simple words, it spreads energy out in a less useful form.

Examples of energy changes caused by friction include:

• brake pads heating up when a car slows down
• hands warming when rubbed together
• a meteor heating up as it moves through the atmosphere because of drag

9. Comparing static friction, kinetic friction, and drag

• Static friction: acts when surfaces are not sliding; prevents motion from starting
• Kinetic friction: acts when surfaces slide past each other
• Drag: acts when an object moves through air or water

Another important difference is this:

• Static and kinetic friction happen between solid surfaces in contact.
• Drag happens when moving through a fluid.

10. Worked Example 1: Finding maximum static friction

A \(10\,\text{kg}\) box rests on a flat floor. The coefficient of static friction is \(\mu_s = 0.40\). What is the maximum static friction?

Step 1: Find the normal force.

On a flat surface:

$$N = mg = (10)(9.8) = 98\,\text{N}$$

Step 2: Use the static friction formula.

$$f_s^{\text{max}} = \mu_s N = (0.40)(98) = 39.2\,\text{N}$$

Answer: The maximum static friction is \(39.2\,\text{N}\).

This means:

• if you push with less than \(39.2\,\text{N}\), the box will not move
• if you push with more than \(39.2\,\text{N}\), the box will start moving

11. Worked Example 2: Static friction while the object stays still

Suppose you push the same box with a force of \(25\,\text{N}\). What is the static friction force?

From Example 1, the maximum static friction is \(39.2\,\text{N}\).

Because \(25\,\text{N}\) is less than \(39.2\,\text{N}\), the box does not move. Static friction adjusts to match your push.

So the static friction force is:

$$f_s = 25\,\text{N}$$

Answer: The static friction is \(25\,\text{N}\), acting opposite your push.

This example shows an important idea: static friction is not always equal to \(\mu_s N\). It can be any value from zero up to its maximum.

12. Worked Example 3: Finding kinetic friction

A \(10\,\text{kg}\) box is now sliding across the same floor. The coefficient of kinetic friction is \(\mu_k = 0.30\). Find the kinetic friction force.

Step 1: Find the normal force.

$$N = mg = (10)(9.8) = 98\,\text{N}$$

Step 2: Use the kinetic friction formula.

$$f_k = \mu_k N = (0.30)(98) = 29.4\,\text{N}$$

Answer: The kinetic friction force is \(29.4\,\text{N}\).

Notice that this is less than the maximum static friction from Example 1. That is common.

13. Worked Example 4: Net force with kinetic friction

A student pulls the sliding box with a horizontal force of \(50\,\text{N}\). The kinetic friction is \(29.4\,\text{N}\). What is the net force, and will the box speed up or slow down?

Step 1: Identify the forces.

• Pulling force: \(50\,\text{N}\) forward
• Kinetic friction: \(29.4\,\text{N}\) backward

Step 2: Find the net force.

$$F_{\text{net}} = 50 - 29.4 = 20.6\,\text{N}$$

Step 3: Interpret the result.

The net force is forward, so the box will accelerate forward. That means it will speed up.

Answer: The net force is \(20.6\,\text{N}\) forward, so the box speeds up.

14. Everyday examples of friction

• Walking: Static friction between your shoes and the ground prevents slipping.
• Writing: Friction between a pencil and paper leaves marks.
• Sliding down a playground slide: Kinetic friction slows you down.
• Parachutes: Drag from air helps slow a falling person.
• Car brakes: Friction between brake pads and wheels reduces motion.

15. How to reduce or increase friction

Sometimes friction is helpful, and sometimes it is not. Engineers and scientists often try to control it.

Ways to reduce friction:

• use lubricants such as oil
• make surfaces smoother
• use wheels or ball bearings
• design streamlined shapes to reduce drag

Ways to increase friction:

• add tread to shoes or tires
• make surfaces rougher
• increase the force pressing surfaces together

16. Common mistakes to avoid

• Mistake 1: Thinking friction always means an object is moving. Static friction acts even when an object stays still.
• Mistake 2: Thinking static friction is always equal to \(\mu_s N\). That value is only the maximum.
• Mistake 3: Forgetting that friction acts opposite the motion or attempted motion.
• Mistake 4: Mixing up kinetic friction and drag. Kinetic friction is between sliding solid surfaces, while drag is from moving through a fluid.
• Mistake 5: Forgetting that friction changes kinetic energy into thermal energy.

17. Quick check for understanding

1. If you push a chair and it does not move, what type of friction is acting?
2. Why is it often harder to start moving a heavy box than to keep it moving?
3. What happens to the friction force if the normal force increases?
4. Why does a parachute increase drag?
5. What form of energy often increases when friction acts?

Answers:

1. Static friction
2. Because maximum static friction is usually greater than kinetic friction
3. Friction usually increases
4. It increases the area interacting with air, which increases air resistance
5. Thermal energy, or heat

Brief Summary

Friction is a force that opposes motion and is caused in part by tiny surface roughness called asperities and by interactions with fluids. Static friction prevents motion from starting, kinetic friction opposes sliding motion, and drag opposes motion through air or water.

The size of static and kinetic friction depends on the normal force and the coefficient of friction. Friction is important because it affects motion, works with Newton’s laws, and transforms some kinetic energy into heat.

Universal Gravitation and the Inverse-Square Law

Gravity is the force that pulls objects toward each other. It is the reason apples fall, the Moon orbits Earth, and planets orbit the Sun. Even though gravity may seem strongest only when we are near Earth, it actually acts between all objects that have mass.

In this lesson, you will learn how Newton described gravity with a mathematical rule called the law of universal gravitation. You will also learn why distance matters so much by studying the inverse-square law.

1. What is universal gravitation?

Isaac Newton explained that every object in the universe pulls on every other object with a gravitational force. This idea is called universal gravitation. The word “universal” means it applies everywhere, not just on Earth.

The size of the gravitational force depends on two main things:

• the masses of the two objects
• the distance between their centers

If the masses are larger, the gravitational force is stronger. If the objects are farther apart, the gravitational force is weaker.

2. Newton’s law of gravitation

Newton wrote the gravitational force using this equation:

In this equation:

• 3F4 = gravitational force
• 3G4 = gravitational constant, a number that stays the same everywhere
• 3m_14 and 3m_24 = the masses of the two objects
• 3r4 = the distance between the centers of the two objects

You do not always need to use the exact value of 3G4 in 9th Grade unless a problem gives it to you. Often, the most important part is understanding how changing mass or distance changes the force.

3. How mass affects gravitational force

The force is directly related to both masses. That means:

• If one mass doubles, the force doubles.
• If both masses double, the force becomes four times as great.
• If one mass is cut in half, the force is cut in half.

This happens because the masses are multiplied together in the formula: \(m_1 m_2\).

4. How distance affects gravitational force

Distance has an even bigger effect because it is squared in the denominator:

This means gravity follows an inverse-square law. “Inverse” means the force gets smaller as distance gets larger. “Square” means you square the distance first.

Here is what happens when distance changes:

• If distance doubles, force becomes \(\frac{1}{2^2} = \frac{1}{4}\) as large.
• If distance triples, force becomes \(\frac{1}{3^2} = \frac{1}{9}\) as large.
• If distance is cut in half, force becomes \(\frac{1}{(1/2)^2} = 4\) times as large.

This is why gravity weakens quickly as objects move farther apart.

5. Why do we use center-to-center distance?

When working with planets, moons, or stars, the distance in the formula is measured from the center of one object to the center of the other. That gives the correct value of \(r\).

For example, when we talk about the gravitational pull between Earth and the Moon, we do not measure from Earth’s surface to the Moon’s surface. We measure from Earth’s center to the Moon’s center.

6. Gravity and orbits

Gravity is what keeps planets in orbit around the Sun and moons in orbit around planets. A planet does not fall straight into the Sun because it is also moving sideways. Gravity continuously pulls it inward, creating a curved path.

Without gravity, planets and moons would move in straight lines. Gravity changes their motion and keeps the solar system together.

7. Gravitational force pairs

If Earth pulls on the Moon, the Moon also pulls on Earth. These forces are equal in size and opposite in direction. This matches Newton’s third law: every action force has an equal and opposite reaction force.

Even though the forces are equal, the motion of the objects may not be the same. A smaller object usually changes motion more than a larger one because it has less mass.

8. Relationship between weight and gravity

Your weight is the gravitational force pulling you toward a planet or moon. Your mass is the amount of matter in your body and does not change when you move to a different place.

For example, your mass would be the same on Earth and the Moon, but your weight would be smaller on the Moon because the Moon has less mass than Earth and pulls with less gravitational force.

9. Worked Example 1: Changing one mass

Suppose two objects attract each other with a force of 20 N. If one object’s mass doubles and the other mass and distance stay the same, what happens to the force?

Step 1: Look at the formula \(F = G\frac{m_1 m_2}{r^2}\).

Step 2: Only one mass changes, and it doubles.

Step 3: Since force is directly proportional to mass, the force also doubles.

Answer: The new force is \(40\text{ N}\).

Worked Example 2: Changing distance

Two planets attract each other with a force of 900 N. If the distance between their centers becomes 3 times greater, what is the new force?

Step 1: Use the inverse-square idea.

If distance is multiplied by 3, force is divided by \(3^2 = 9\).

Step 2: Calculate the new force:

Answer: The new gravitational force is \(100\text{ N}\).

Worked Example 3: Changing both masses

A gravitational force between two objects is 50 N. If one mass doubles and the other mass triples, while the distance stays the same, what is the new force?

Step 1: Multiply the mass changes together.

Doubling one mass gives a factor of 2. Tripling the other gives a factor of 3.

Step 2: Total change in force:

Step 3: Multiply the original force by 6.

Answer: The new force is \(300\text{ N}\).

Worked Example 4: Comparing two distances

An asteroid experiences a gravitational force of 16 N at a certain distance from a planet. If the asteroid moves to half that distance, what happens to the force?

Step 1: Half the distance means \(r\) becomes \(\frac{1}{2}r\).

Step 2: Because gravity follows an inverse-square law, cutting distance in half makes the force 4 times larger.

Step 3: Calculate the new force:

Answer: The new force is \(64\text{ N}\).

10. Common mistakes to avoid

• Forgetting to square the distance. If distance doubles, the force does not get cut in half. It becomes one-fourth as much.
• Confusing mass and weight. Mass is how much matter an object has. Weight is the gravitational force on that mass.
• Using surface distance instead of center-to-center distance. In space problems, use the distance between centers.
• Thinking gravity only exists near Earth. Gravity acts between all masses everywhere in the universe.

11. Key ideas to remember

• Every mass pulls on every other mass.
• Newton’s law of gravitation is $$F = G\frac{m_1 m_2}{r^2}$$
• More mass means stronger gravitational force.
• More distance means weaker gravitational force.
• Because of the inverse-square law, distance changes have a large effect.
• Gravity keeps moons and planets in orbit.

Brief Summary

Universal gravitation means that all objects with mass attract each other. Newton showed that gravitational force increases with mass and decreases with the square of the distance between objects. The inverse-square law explains why gravity becomes much weaker as distance grows, and this idea helps us understand weight, planetary motion, and orbits in space.

Projectile motion happens when an object is launched into the air and then moves under the pull of gravity. A thrown ball, a kicked soccer ball, or water spraying from a hose can all show projectile motion.

The big idea in this lesson is vector independence. This means the motion in the horizontal direction and the motion in the vertical direction can be studied separately. Even though the object follows one curved path, its sideways motion and up-and-down motion do not affect each other.

Understanding this idea makes projectile motion much easier. Instead of solving one difficult motion problem, we split it into two simpler parts:

• Horizontal motion: constant velocity
• Vertical motion: constant acceleration due to gravity

Let us build this idea step by step.

1. What is a projectile?

A projectile is an object that is moving through the air after it has been launched, and the main force acting on it is gravity. In basic 9th Grade physics, we usually ignore air resistance so that the motion is easier to study.

Once the object leaves the launcher or the thrower’s hand, gravity pulls it downward. Gravity gives the object a downward acceleration of about:

For many school problems, this is rounded to:

2. Why the path is curved

If there were no gravity, a launched object would keep moving in a straight line at constant speed. But gravity pulls it downward every second. Because of this, the object keeps moving forward while also falling downward, so its path becomes a curve.

This curved path is called a parabolic path. You do not need advanced math to understand it. Just remember: the object moves forward and falls at the same time.

3. Vector independence

Velocity is a vector, which means it has both size and direction. In projectile motion, we separate the velocity into:

• Horizontal component — motion across
• Vertical component — motion up or down

These two components act independently:

• The horizontal velocity stays constant if air resistance is ignored.
• The vertical velocity changes because gravity accelerates the object downward.

This means gravity changes only the vertical motion, not the horizontal motion.

For example, imagine dropping one ball straight down and launching another ball horizontally from the same height at the same time. The launched ball moves sideways, but both balls fall downward in the same way. They hit the ground at the same time if air resistance is ignored.

4. Horizontal motion

In the horizontal direction, there is no acceleration in ideal projectile motion:

That means the horizontal velocity stays the same:

So horizontal distance is found using:

where:

• \(d_x\) = horizontal distance
• \(v_x\) = horizontal velocity
• \(t\) = time

5. Vertical motion

In the vertical direction, gravity causes constant acceleration downward:

The negative sign shows that gravity acts downward if we choose upward as positive.

The vertical velocity changes with time:

The vertical position can be found with:

where:

• \(d_y\) = vertical displacement
• \(v_{y0}\) = starting vertical velocity
• \(a_y\) = vertical acceleration
• \(t\) = time

If the projectile is launched horizontally, then:

and the vertical displacement becomes:

6. Launched horizontally vs. launched at an angle

There are two common projectile cases in 9th Grade science.

Case A: Horizontal launch

The object starts with sideways speed only. Its initial vertical velocity is zero.

• Horizontal motion: constant velocity
• Vertical motion: starts from rest and speeds up downward

Case B: Launch at an angle

The object starts with both horizontal and vertical velocity. It moves upward at first if the initial vertical component points upward, but gravity slows the upward motion until the object reaches the highest point. Then it falls back down.

At the highest point:

But the horizontal velocity is still not zero, so the projectile continues moving forward.

7. Breaking a launch velocity into components

If an object is launched at an angle, we split the starting velocity into two parts:

• \(v_x\) = horizontal component
• \(v_y\) = vertical component

You may sometimes be given these components directly. If the total launch speed and angle are given, the components are:

where:

• \(v\) = launch speed
• \(\theta\) = launch angle above the horizontal

You do not always need to calculate these in detail unless your class has practiced it, but it is important to understand that one velocity can be split into sideways and upward parts.

8. Time is the link between the two directions

The horizontal motion and vertical motion are independent, but they share one important thing: time.

The projectile spends the same amount of time moving horizontally and vertically because both motions happen at the same time. In many problems, we first use the vertical motion to find the time in the air, and then use that same time in the horizontal motion equation.

Worked Example 1: Horizontal launch from a table

A ball rolls off a table with a horizontal speed of \(4 \text{ m/s}\). It takes \(0.5 \text{ s}\) to hit the floor. How far from the table does it land?

Step 1: Identify the horizontal information.

• \(v_x = 4 \text{ m/s}\)
• \(t = 0.5 \text{ s}\)

Step 2: Use the horizontal distance formula.

Answer: The ball lands \(2 \text{ m}\) from the table.

What to notice: We did not need gravity to calculate the horizontal distance because the horizontal velocity stays constant.

Worked Example 2: Finding falling time from vertical motion

A rock is launched horizontally from a cliff. It falls \(20 \text{ m}\) downward. How long is it in the air? Use \(g = 10 \text{ m/s}^2\).

Step 1: Use vertical motion.

Since it is launched horizontally:

Use:

The rock falls \(20\) m downward. Using positive downward for simplicity:

Answer: The rock is in the air for \(2 \text{ s}\).

What to notice: The time came from the vertical motion, because gravity controls how long the object takes to fall.

Worked Example 3: Use time to find horizontal range

A ball is launched horizontally at \(6 \text{ m/s}\) from a height of \(45 \text{ m}\). How far does it travel horizontally before hitting the ground? Use \(g = 10 \text{ m/s}^2\).

Step 1: Find the time using vertical motion.

Step 2: Use that time in horizontal motion.

Answer: The ball travels \(18 \text{ m}\) horizontally.

What to notice: First find time from vertical motion, then use the same time for horizontal motion.

Worked Example 4: Projectile launched upward at an angle

A ball is kicked so that its initial velocity components are:

• Horizontal: \(v_x = 8 \text{ m/s}\)
• Vertical: \(v_{y0} = 12 \text{ m/s}\)

Use \(g = 10 \text{ m/s}^2\).

(a) How long does it take to reach the highest point?

At the highest point, the vertical velocity is zero.

Answer: It takes \(1.2 \text{ s}\) to reach the highest point.

(b) How far does it travel horizontally in that time?

Answer: It travels \(9.6 \text{ m}\) horizontally by the time it reaches the highest point.

What to notice: Even at the highest point, the horizontal motion continues. Only the vertical velocity becomes zero for an instant.

9. Important ideas to remember

• Horizontal and vertical motion are independent.
• Gravity affects only the vertical motion.
• Horizontal velocity stays constant if air resistance is ignored.
• Vertical velocity changes because of gravity.
• Time connects both directions.
• The path of a projectile is curved.

10. Common mistakes

Students often mix the horizontal and vertical parts of the motion. Watch out for these errors:

• Thinking gravity slows horizontal motion. In ideal projectile motion, it does not.
• Using vertical equations for horizontal distance. Horizontal motion uses constant velocity: \(d_x = v_x t\).
• Forgetting that the same time applies to both directions.
• Thinking the object stops completely at the top. Only the vertical velocity is zero there; horizontal velocity remains.
• Mixing up speed and velocity. Velocity includes direction.

11. Quick comparison table

• Horizontal direction:
  • Acceleration: \(0\)
  • Velocity: constant
  • Main equation: \(d_x = v_x t\)
• Vertical direction:
  • Acceleration: \(-9.8 \text{ m/s}^2\) or about \(-10 \text{ m/s}^2\)
  • Velocity: changes over time
  • Main equations: \(v_y = v_{y0} + a_y t\), \(d_y = v_{y0} t + \frac{1}{2} a_y t^2\)

12. Real-world examples

You can see projectile motion in many everyday situations:

• A basketball shot toward the hoop
• A soccer ball kicked across a field
• A ball rolling off a desk
• Water arcing from a fountain

In each case, the object moves forward while gravity pulls it downward.

Brief Summary

Projectile motion is the motion of an object moving through the air under the influence of gravity. The key idea is vector independence: horizontal motion and vertical motion can be treated separately. Horizontally, the velocity stays constant. Vertically, the object accelerates downward because of gravity. The same time is used in both directions, which helps us solve projectile problems step by step.

Momentum and Impulse are two ideas that help us understand how motion changes when forces act on objects.

Have you ever noticed that a moving shopping cart is harder to stop than a rolling ball? Or that a catcher pulls a glove backward when catching a fast baseball? These are examples of momentum and impulse in action.

In this lesson, you will learn what momentum is, how to calculate it, and how a force acting over time can change an object's motion.

1. What is momentum?

Momentum is the amount of motion an object has. An object with more mass or more speed has more momentum.

The formula for momentum is:

$$p = m \times v$$

In this formula:

• p = momentum
• m = mass
• v = velocity

Velocity means speed in a certain direction. For many basic problems, you can think of it as speed, but direction still matters.

This means:

• If mass increases, momentum increases.
• If velocity increases, momentum increases.
• If an object is not moving, its velocity is 0, so its momentum is 0.

For example, a heavy truck moving slowly can have a lot of momentum. A light tennis ball moving very fast can also have momentum, but usually less than the truck.

2. What is impulse?

Impulse is the effect of a force acting over a period of time. Impulse changes an object's momentum.

The formula for impulse is:

$$J = F \times t$$

In this formula:

• J = impulse
• F = force
• t = time

A bigger force gives a bigger impulse. A longer time also gives a bigger impulse.

Impulse and change in momentum are connected by this idea:

$$F \times t = \text{change in momentum}$$

This means that when you apply a force for some amount of time, you change how much momentum an object has.

3. Why time matters

Sometimes we want to change momentum without using a huge force. One way to do that is to increase the time over which the force acts.

That is why many safety tools are designed to make collisions happen over a longer time.

• Seat belts help slow your body down over a little more time.
• Air bags spread out the force and increase the stopping time.
• Helmets help reduce the force on your head by increasing the time of impact.
• Gym mats make stopping take longer when you land.

When the time becomes longer, the force can be smaller for the same change in momentum.

4. Momentum can increase, decrease, or change direction

Because momentum depends on velocity, momentum can change in different ways:

• An object speeds up, so momentum increases.
• An object slows down, so momentum decreases.
• An object changes direction, so momentum changes even if speed stays the same.

For example, when a soccer ball is kicked back in the opposite direction, its momentum changes a lot because the direction changed.

5. Worked Example 1: Finding momentum

A toy car has a mass of 2 kg and moves at 3 m/s. What is its momentum?

Use the formula:

$$p = m \times v$$

Substitute the values:

$$p = 2 \times 3$$

$$p = 6$$

The momentum is 6 kg·m/s.

Worked Example 2: Comparing momentum

Which has more momentum?

• A 4 kg ball moving at 2 m/s
• A 2 kg ball moving at 5 m/s

First object:

$$p = 4 \times 2 = 8$$

Momentum = 8 kg·m/s

Second object:

$$p = 2 \times 5 = 10$$

Momentum = 10 kg·m/s

The 2 kg ball moving at 5 m/s has more momentum.

Worked Example 3: Finding impulse

A force of 6 N acts on a skateboard for 4 seconds. What is the impulse?

Use the formula:

$$J = F \times t$$

Substitute the values:

$$J = 6 \times 4$$

$$J = 24$$

The impulse is 24 N·s.

This also means the skateboard's momentum changed by 24.

Worked Example 4: How force and time work together

Two players stop identical rolling balls.

• Player A uses a force of 10 N for 2 s.
• Player B uses a force of 5 N for 4 s.

Find the impulse for each player.

Player A:

$$J = 10 \times 2 = 20$$

Player B:

$$J = 5 \times 4 = 20$$

Both players give an impulse of 20 N·s.

This means both change the ball's momentum by the same amount, even though one used a bigger force for less time and the other used a smaller force for more time.

6. Real-life examples

• Kicking a ball: Your foot applies a force for a short time, giving the ball impulse and increasing its momentum.
• Catching a ball: Moving your hands backward increases the time of impact, which reduces the force.
• Hammering a nail: The hammer has momentum as it swings. When it hits the nail, the force acts over a short time and changes the nail's motion.
• Car safety features: Seat belts and air bags increase stopping time, reducing force on passengers.

7. Common mistakes to avoid

• Mixing up mass and weight: In momentum problems, use mass.
• Forgetting time in impulse: Impulse depends on both force and time.
• Thinking only heavy objects have momentum: Light objects can also have momentum if they move fast.
• Forgetting direction matters: A change in direction means a change in momentum.

8. Key ideas to remember

• Momentum is the amount of motion an object has.
• Momentum depends on mass and velocity.
• The formula for momentum is \(p = m \times v\).
• Impulse is force acting over time.
• The formula for impulse is \(J = F \times t\).
• Impulse changes momentum.
• A longer stopping time can reduce force.

Summary

Momentum tells us how much motion an object has, and it depends on mass and velocity. Impulse happens when a force acts over time, and that impulse changes momentum.

By understanding momentum and impulse, you can explain why bigger, faster objects are harder to stop and why safety tools like helmets, mats, and air bags are so important.

Uniform Circular Motion and Centripetal Force

Have you ever watched a car go around a roundabout, a rider on a Ferris wheel, or a ball being swung on a string? In all of these situations, the object is moving in a circle. Even if the object’s speed stays the same, its motion is still changing because its direction keeps changing.

This kind of motion is called uniform circular motion. The word uniform means constant, so uniform circular motion means motion in a circle at a constant speed.

In this lesson, you will learn why an object moving in a circle is still accelerating, what centripetal force is, and how to calculate it.

1. What is uniform circular motion?

An object is in uniform circular motion when:

• it moves along a circular path, and
• its speed remains constant.

Even though the speed does not change, the velocity changes. This is because velocity includes both speed and direction.

For example, if a runner moves around a circular track at a steady speed, the runner is always turning. That means the direction of motion changes at every moment. Since velocity is changing, the runner is accelerating.

2. Why is there acceleration if the speed stays the same?

Acceleration means a change in velocity. Velocity changes when either:

• speed changes, or
• direction changes.

In uniform circular motion, the speed stays the same, but the direction changes continuously. So the object has an acceleration.

This acceleration points toward the center of the circle. It is called centripetal acceleration.

The word centripetal means center-seeking. That helps us remember that the acceleration always points inward, toward the center of the circular path.

3. What is centripetal force?

If an object is accelerating, then there must be a net force acting on it. In circular motion, this force must also point toward the center of the circle.

This inward force is called centripetal force.

Centripetal force is not a new kind of force. It is just the name for whatever force is pulling or pushing the object toward the center. Depending on the situation, the centripetal force could be:

• tension in a string,
• gravity for planets and satellites,
• friction for a car turning on a road,
• or the normal force in some curved motions.

Without this inward force, the object would not keep moving in a circle. Instead, it would move off in a straight line.

4. Direction of motion and force

In circular motion:

• the object’s velocity points along the circle, in the direction it is moving,
• the centripetal force points toward the center.

This means the velocity and the centripetal force point in different directions.

A helpful way to picture this is to imagine a ball tied to a string and swung in a circle. The ball’s motion is along the circular path, but the string pulls the ball inward toward your hand. That inward pull is the centripetal force.

5. Formula for centripetal force

The formula for centripetal force is:

Where:

• \(F_c\) = centripetal force in newtons (N)
• \(m\) = mass in kilograms (kg)
• \(v\) = speed in meters per second (m/s)
• \(r\) = radius of the circle in meters (m)

This formula shows some important ideas:

• If mass increases, centripetal force increases.
• If speed increases, centripetal force increases a lot, because speed is squared.
• If radius increases, centripetal force decreases.

6. Formula for centripetal acceleration

The acceleration of an object in uniform circular motion is:

Where:

• \(a_c\) = centripetal acceleration in meters per second squared \((m/s^2)\)
• \(v\) = speed
• \(r\) = radius

This also points toward the center of the circle.

Notice that Newton’s second law connects the two formulas:

If we use centripetal acceleration, then:

7. What happens if the centripetal force disappears?

Sometimes students think an object “wants” to keep moving in a circle. That is not correct.

According to Newton’s first law, an object keeps moving in a straight line unless a force changes its motion. In circular motion, the inward force keeps changing the direction.

If the centripetal force suddenly disappears, the object will move in a straight line tangent to the circle. A tangent is a straight line that touches the circle at one point.

For example, if a string breaks while a ball is being swung, the ball flies off in the direction it was moving at that moment, not toward the outside in a curved path.

8. Real-life examples of centripetal force

• Car turning on a curve: Friction between the tires and the road provides the centripetal force.
• Moon orbiting Earth: Gravity provides the centripetal force.
• Ball on a string: Tension in the string provides the centripetal force.
• Roller coaster loop: The track pushes on the car to help provide centripetal force.

9. Worked Examples

Example 1: Finding centripetal acceleration

A toy car moves in a circle of radius \(2\,m\) at a constant speed of \(4\,m/s\). Find its centripetal acceleration.

Step 1: Use the formula

Step 2: Substitute values

Answer: The centripetal acceleration is \(8\,m/s^2\), directed toward the center of the circle.

Example 2: Finding centripetal force

A \(0.5\,kg\) ball is swung in a circle of radius \(1.5\,m\) at a speed of \(3\,m/s\). Find the centripetal force.

Step 1: Use the formula

Step 2: Substitute values

Answer: The centripetal force is \(3\,N\), toward the center.

Example 3: Comparing speeds

A cyclist rides in a circle of radius \(10\,m\). If the cyclist doubles speed from \(2\,m/s\) to \(4\,m/s\), how does the centripetal force change, assuming mass stays the same?

Use the relationship:

If speed doubles, then:

Answer: The centripetal force becomes 4 times larger.

This is an important result. A small increase in speed can cause a much bigger increase in the force needed to keep an object moving in a circle.

Example 4: Solving for speed

A \(2\,kg\) object moves in a circle of radius \(8\,m\). The centripetal force is \(18\,N\). Find the speed.

Step 1: Start with the formula

Step 2: Rearrange to solve for \(v^2\)

Step 3: Substitute values

Step 4: Take the square root

Answer: The speed is about \(8.5\,m/s\).

10. Common mistakes to avoid

• Thinking no acceleration exists because speed is constant: Acceleration can happen when direction changes.
• Thinking centripetal force pushes outward: Centripetal force always points inward.
• Forgetting that speed is squared: In \(F_c = \frac{mv^2}{r}\), speed has a very strong effect.
• Mixing up velocity and force directions: Velocity is tangent to the circle, while centripetal force points toward the center.

11. Key ideas to remember

• Uniform circular motion means moving in a circle at constant speed.
• An object in circular motion is accelerating because its direction changes.
• The acceleration is called centripetal acceleration and points toward the center.
• The force causing this motion is called centripetal force.
• The centripetal force formula is \(F_c = \frac{mv^2}{r}\).
• If the inward force stops, the object moves in a straight line tangent to the circle.

Brief Summary

Uniform circular motion happens when an object moves in a circle at constant speed. Even though the speed stays the same, the object is still accelerating because its direction keeps changing. This acceleration, and the force causing it, always point toward the center of the circle. Understanding centripetal force helps explain turning cars, orbiting moons, swinging objects, and many other kinds of motion in everyday life.

Linear Momentum and Impulse

When objects move, they do not all behave the same way. A slowly rolling tennis ball is easy to stop, but a fast-moving bicycle is much harder to stop. This difference is explained by a quantity called momentum.

Momentum helps us describe how hard it is to stop a moving object. It depends on two things: the object's mass and its velocity. In this lesson, you will learn how to calculate momentum, what impulse means, and how these ideas help explain safety features such as airbags and seat belts.

1. What is linear momentum?

Linear momentum is the amount of motion an object has in a straight line. The symbol for momentum is usually \(p\).

The formula for momentum is:

$$p = mv$$

where:

• \(p\) = momentum
• \(m\) = mass in kilograms (kg)
• \(v\) = velocity in meters per second (m/s)

The unit for momentum is:

$$\text{kg·m/s}$$

Because velocity includes direction, momentum also has direction. That means two objects with the same mass and speed can have different momentum if they move in different directions.

For example, if we say motion to the right is positive, then motion to the left would be negative.

2. How mass and velocity affect momentum

An object has more momentum if:

• its mass is larger,
• its velocity is greater,
• or both.

A heavy truck moving slowly can have a lot of momentum. A light ball moving very fast can also have momentum, but usually less than the truck.

If an object is not moving, then its velocity is 0. That means its momentum is also 0.

$$p = m(0) = 0$$

3. What is impulse?

Sometimes an object's momentum changes. To change momentum, a force must act on the object for a period of time. This effect is called impulse.

Impulse depends on:

• the size of the force, and
• how long the force acts.

The formula for impulse is:

$$J = F\Delta t$$

where:

• \(J\) = impulse
• \(F\) = force in newtons (N)
• \(\Delta t\) = time interval in seconds (s)

The unit for impulse is:

$$\text{N·s}$$

This unit is equal to \(\text{kg·m/s}\), which is the same as momentum.

4. The impulse-momentum theorem

The most important idea in this lesson is that impulse equals the change in momentum.

$$J = \Delta p$$

This can also be written as:

$$F\Delta t = mv_f - mv_i$$

where:

• \(v_i\) = initial velocity
• \(v_f\) = final velocity

This means that if you know the force and the time, you can find how much the momentum changes. Or, if you know the change in momentum and the time, you can find the force.

5. Why increasing time can reduce force

This idea is very important in crash safety. In a collision, a person's momentum must change very quickly. If that change happens in a very short time, the force is large.

From the formula

$$F\Delta t = \Delta p$$

if \(\Delta p\) stays the same, then making \(\Delta t\) larger makes \(F\) smaller.

That is why safety devices are designed to increase the stopping time.

• Seat belts stretch slightly and help stop a person over a longer time.
• Airbags cushion the impact and increase the time it takes to stop.
• Crash mats and helmets also increase stopping time.

These devices do not remove momentum by magic. They reduce the force by spreading the change in momentum over a longer time.

6. Direction matters

Since momentum depends on velocity, direction matters. If an object changes direction, its momentum changes even if its speed stays the same.

For example:

• A ball moving at \(+4\,\text{m/s}\) has positive momentum.
• If it bounces back at \(-4\,\text{m/s}\), the final momentum is negative.

This is a large change in momentum because the direction reversed.

7. Worked Example 1: Finding momentum

A \(3\,\text{kg}\) cart moves at \(5\,\text{m/s}\) to the right. Find its momentum.

Step 1: Use the formula

$$p = mv$$

Step 2: Substitute the values

$$p = (3)(5)$$

Step 3: Calculate

$$p = 15\,\text{kg·m/s}$$

Answer: The cart's momentum is \(15\,\text{kg·m/s}\) to the right.

8. Worked Example 2: Finding impulse

A force of \(12\,\text{N}\) acts on a ball for \(0.5\,\text{s}\). What is the impulse?

Step 1: Use the formula

$$J = F\Delta t$$

Step 2: Substitute the values

$$J = (12)(0.5)$$

Step 3: Calculate

$$J = 6\,\text{N·s}$$

Answer: The impulse is \(6\,\text{N·s}\).

Since impulse equals change in momentum, the ball's momentum changed by \(6\,\text{kg·m/s}\).

9. Worked Example 3: Using impulse to find final velocity

A \(2\,\text{kg}\) soccer ball is initially at rest. A player kicks it with an impulse of \(8\,\text{N·s}\). What is the ball's final velocity?

Step 1: Use the impulse-momentum theorem

$$J = \Delta p = mv_f - mv_i$$

The ball starts at rest, so \(v_i = 0\).

$$8 = 2v_f - 2(0)$$

Step 2: Simplify

$$8 = 2v_f$$

Step 3: Solve for \(v_f\)

$$v_f = 4\,\text{m/s}$$

Answer: The ball's final velocity is \(4\,\text{m/s}\) in the direction of the kick.

10. Worked Example 4: Crash safety and force

A \(60\,\text{kg}\) passenger in a car is moving at \(20\,\text{m/s}\). During a crash, the passenger comes to rest.

First, find the change in momentum:

$$\Delta p = mv_f - mv_i$$

$$\Delta p = 60(0) - 60(20)$$

$$\Delta p = -1200\,\text{kg·m/s}$$

The negative sign shows the momentum decreased. The size of the change is \(1200\,\text{kg·m/s}\).

Case A: Stopping time is \(0.10\,\text{s}\)

$$F = \frac{\Delta p}{\Delta t}$$

$$F = \frac{1200}{0.10} = 12000\,\text{N}$$

Case B: Stopping time is \(0.50\,\text{s}\)

$$F = \frac{1200}{0.50} = 2400\,\text{N}$$

Answer: When the stopping time increases from \(0.10\,\text{s}\) to \(0.50\,\text{s}\), the force becomes much smaller.

This is why airbags and seat belts are so important: they increase the time of impact and reduce the force on the body.

11. Common mistakes to avoid

• Confusing speed and velocity: Momentum uses velocity, so direction matters.
• Forgetting units: Momentum is in \(\text{kg·m/s}\), and impulse is in \(\text{N·s}\).
• Ignoring signs: Positive and negative directions help show changes in motion.
• Mixing up mass and weight: Use mass in kilograms, not weight in newtons.
• Forgetting that impulse changes momentum: Impulse is not the same as momentum, but it tells how much momentum changes.

12. Key ideas to remember

• Momentum is the product of mass and velocity.
• $$p = mv$$
• Impulse is force multiplied by time.
• $$J = F\Delta t$$
• Impulse equals change in momentum.
• $$J = \Delta p$$
• Increasing the time of a collision reduces the force if the change in momentum stays the same.
• Crash safety devices work by increasing stopping time.

Brief Summary

Linear momentum measures how much motion an object has and is found using \(p = mv\). Impulse measures the effect of a force acting over time and is found using \(J = F\Delta t\). The impulse-momentum theorem, \(J = \Delta p\), connects these ideas and helps explain real-life situations like sports, car crashes, airbags, and seat belts. The longer the stopping time, the smaller the force during a collision.

Conservation of Momentum in Collisions

When two objects collide, their motion can change very quickly. A fast object may slow down, a slow object may speed up, or two objects may stick together after crashing. Even though their individual speeds change, there is an important rule that helps us predict what happens: the total momentum of the system stays the same if no outside forces act on it.

This idea is called conservation of momentum. It is one of the most useful tools in mechanics because it lets us figure out what happens after a collision by using what we know before the collision.

In this lesson, you will learn what momentum is, how to use conservation of momentum, and how collisions differ when objects bounce apart or stick together.

1. What is momentum?

Momentum is a measure of how much motion an object has. It depends on two things:

• the object's mass
• the object's velocity

The formula for momentum is:

where:

• \(p\) = momentum

• \(m\) = mass

• \(v\) = velocity

Because velocity includes direction, momentum also includes direction. This means momentum can be positive or negative.

For example, if we choose motion to the right as positive:

• a cart moving right has positive momentum
• a cart moving left has negative momentum

2. What does conservation mean?

When we say momentum is conserved, we mean the total momentum before a collision equals the total momentum after the collision, as long as the system is isolated.

An isolated system is a system where outside forces do not affect the total momentum in an important way during the collision. In many short collisions, we can ignore outside forces like friction because the collision happens so quickly.

The conservation of momentum equation is:

For two objects, this can be written as:

Here:

• \(m_1, m_2\) are the masses
• \(v_1, v_2\) are the velocities before the collision
• \(v_1', v_2'\) are the velocities after the collision

3. Why direction matters

Momentum is not just about speed. Direction is part of velocity, so it must be included in every momentum problem.

Before solving a collision problem, choose a positive direction. Then:

• motion in the positive direction gets a positive velocity
• motion in the opposite direction gets a negative velocity

This helps the math correctly show whether momenta add or cancel.

4. Types of collisions

There are different kinds of collisions, but in all of them, momentum is conserved if the system is isolated.

A. Perfectly inelastic collision

In a perfectly inelastic collision, the objects stick together after they collide and move as one object.

Since they move together after the collision, they have the same final velocity, which we can call \(v_f\).

The equation becomes:

This is often the easiest type of collision to solve because there is only one final velocity.

B. Perfectly elastic collision

In a perfectly elastic collision, the objects bounce off each other without losing kinetic energy. For 9th Grade, the most important idea is that momentum is conserved, and the objects do not stick together.

That means after the collision, each object may have a different final velocity:

In many problems, one of the final velocities is given, and you solve for the other using momentum conservation.

5. Steps for solving momentum collision problems

1. List the masses and velocities of all objects.

2. Choose a positive direction.

3. Give each velocity the correct sign.

4. Write the conservation of momentum equation.

5. Solve for the unknown value.

6. Check whether your answer makes sense.

6. Worked Example 1: One moving object hits a stationary object and they stick together

A \(2\,\text{kg}\) cart moving to the right at \(4\,\text{m/s}\) collides with a \(3\,\text{kg}\) cart at rest. The carts stick together. Find their final velocity.

Step 1: Write what is known.

• \(m_1 = 2\,\text{kg}\)
• \(v_1 = 4\,\text{m/s}\)
• \(m_2 = 3\,\text{kg}\)
• \(v_2 = 0\,\text{m/s}\)

Step 2: Use the perfectly inelastic collision formula.

Substitute the values:

Answer: The carts move together at \(1.6\,\text{m/s}\) to the right.

This result makes sense because the moving cart shares its momentum with the heavier cart, so the final speed is smaller than \(4\,\text{m/s}\).

7. Worked Example 2: Two objects moving toward each other and sticking

A \(1\,\text{kg}\) ball moves right at \(6\,\text{m/s}\). A \(2\,\text{kg}\) ball moves left at \(2\,\text{m/s}\). They collide and stick together. Find their final velocity.

Step 1: Choose right as positive.

• \(m_1 = 1\,\text{kg}\)
• \(v_1 = +6\,\text{m/s}\)
• \(m_2 = 2\,\text{kg}\)
• \(v_2 = -2\,\text{m/s}\)

Step 2: Use conservation of momentum.

Answer: The stuck-together balls move at about \(0.67\,\text{m/s}\) to the right.

Notice that the second ball had negative momentum because it was moving left. Its momentum partly canceled the first ball's momentum.

8. Worked Example 3: Elastic collision with one final velocity given

A \(2\,\text{kg}\) cart moves right at \(5\,\text{m/s}\) and collides with a \(1\,\text{kg}\) cart at rest. After the collision, the \(2\,\text{kg}\) cart moves right at \(2\,\text{m/s}\). Find the final velocity of the \(1\,\text{kg}\) cart.

Step 1: Write the known values.

• \(m_1 = 2\,\text{kg}\)
• \(v_1 = +5\,\text{m/s}\)
• \(m_2 = 1\,\text{kg}\)
• \(v_2 = 0\,\text{m/s}\)
• \(v_1' = +2\,\text{m/s}\)
• \(v_2' = ?\)

Step 2: Use conservation of momentum.

Answer: The \(1\,\text{kg}\) cart moves right at \(6\,\text{m/s}\) after the collision.

This shows that in an elastic collision, the objects do not move together. Each object can leave with its own speed.

9. Worked Example 4: Collision with opposite directions after impact

A \(3\,\text{kg}\) object moves right at \(4\,\text{m/s}\). It collides with a \(2\,\text{kg}\) object moving right at \(1\,\text{m/s}\). After the collision, the \(3\,\text{kg}\) object moves left at \(1\,\text{m/s}\). Find the final velocity of the \(2\,\text{kg}\) object.

Step 1: Choose right as positive.

• \(m_1 = 3\,\text{kg}\)
• \(v_1 = +4\,\text{m/s}\)
• \(m_2 = 2\,\text{kg}\)
• \(v_2 = +1\,\text{m/s}\)
• \(v_1' = -1\,\text{m/s}\)
• \(v_2' = ?\)

Step 2: Apply conservation of momentum.

Answer: The \(2\,\text{kg}\) object moves right at \(8.5\,\text{m/s}\).

The answer is positive, so it moves to the right. Its speed becomes quite large because it gains momentum while the other object reverses direction.

10. Common mistakes to avoid

• Forgetting direction: Velocities to the left or backward should be negative if right or forward is positive.

• Using speed instead of velocity: Speed has no sign, but momentum needs direction.

• Mixing up before and after values: Keep initial and final quantities clearly labeled.

• Forgetting that stuck objects share one final velocity: In a perfectly inelastic collision, both objects move together after impact.

• Adding masses incorrectly: Only add masses together when the objects stick together and move as one object after the collision.

11. How momentum connects to real life

Conservation of momentum helps explain many real events:

• bumper cars colliding at an amusement park
• a cue ball hitting another ball in pool
• shopping carts crashing together
• car crashes, where vehicles may bounce apart or crumple together

In each case, total momentum before the collision matches total momentum after the collision, as long as the system is treated as isolated during the short collision time.

12. Quick problem-solving checklist

• Did I choose a positive direction?

• Did I give each velocity the correct sign?

• Did I write total momentum before equal to total momentum after?

• If the objects stick together, did I use one shared final velocity?

• Does my final answer include both magnitude and direction?

Summary

Momentum is found using \(p = mv\), and because velocity has direction, momentum has direction too. In an isolated system, the total momentum before a collision equals the total momentum after the collision.

In a perfectly inelastic collision, objects stick together and share one final velocity. In a perfectly elastic collision, objects bounce apart, but momentum is still conserved.

If you carefully track mass, velocity, and direction, you can predict what happens after many kinds of collisions.

Mechanical Work and Power are two important ideas in science that help us describe how forces transfer energy and how fast that transfer happens.

You may use force every day when you push a door, lift a backpack, or ride a bicycle. But in science, not every force means work is being done. Mechanical work happens only when a force causes an object to move in the direction of the force.

Power tells us how quickly work is done. Two people may do the same amount of work, but the one who finishes faster uses more power.

In this lesson, you will learn what mechanical work means, how to calculate it, when work is zero, and how to calculate power.

1. What is Mechanical Work?

In science, work is the transfer of energy when a force moves an object through a distance.

The formula for work is:

$$W = F \times d$$

where:

• W = work
• F = force
• d = distance moved in the direction of the force

The unit of work is the joule (J).

$$1\text{ joule} = 1\text{ newton} \times 1\text{ meter}$$

This means if a force of 1 newton moves an object 1 meter in the direction of the force, 1 joule of work is done.

2. When Is Work Done?

Work is done only if both of these happen:

• A force acts on an object.
• The object moves because of that force.

For example, if you push a box and it slides across the floor, you do work on the box.

But if you push very hard on a wall and the wall does not move, then:

$$W = F \times 0 = 0$$

So no mechanical work is done on the wall, even though you may feel tired.

3. Work Depends on Direction

The distance in the formula must be in the same direction as the force.

If the force and motion are in the same direction, work is positive.

If the force and motion are in opposite directions, work is negative.

If the force is sideways to the motion, then that force does no work.

A simple 9th Grade way to think about this is:

• Positive work: the force helps the motion.
• Negative work: the force opposes the motion.
• Zero work: the force does not affect motion in that direction.

For example:

• Pushing a shopping cart forward: positive work
• Friction slowing a sliding book: negative work
• Carrying a book across a room at constant height: your upward force is not in the same direction as the horizontal motion, so that upward force does zero work on the book

4. Work and Energy

Mechanical work is closely connected to energy transfer. When work is done on an object, energy is transferred to or from that object.

For example:

• Lifting a book gives it more gravitational potential energy.
• Pushing a cart can increase its kinetic energy.
• Friction can remove mechanical energy by changing it into heat.

So when you calculate work, you are often also describing a change in energy.

5. Calculating Work

To calculate work, follow these steps:

1. Find the force in newtons (N).
2. Find the distance moved in meters (m).
3. Make sure the distance is in the direction of the force.
4. Multiply force by distance.

$$W = F \times d$$

Worked Example 1: Basic Work

A student pushes a box with a force of 20 N across the floor for 5 m. How much work is done?

Step 1: Write the formula.

$$W = F \times d$$

Step 2: Substitute the values.

$$W = 20 \times 5$$

Step 3: Calculate.

$$W = 100\text{ J}$$

Answer: The student does 100 J of work on the box.

Worked Example 2: No Movement

A person pushes on a heavy table with a force of 150 N, but the table does not move. How much work is done on the table?

Step 1: Use the formula.

$$W = F \times d$$

Step 2: Since the table does not move, the distance is 0 m.

$$W = 150 \times 0$$

Step 3: Calculate.

$$W = 0\text{ J}$$

Answer: 0 J of work is done on the table.

6. What is Power?

Power is the rate at which work is done or energy is transferred.

The formula for power is:

$$P = \frac{W}{t}$$

where:

• P = power
• W = work
• t = time

The unit of power is the watt (W).

$$1\text{ watt} = 1\text{ joule per second}$$

If something does 50 joules of work in 10 seconds, its power is 5 watts.

7. Why Power Matters

Power tells us how quickly energy changes happen.

Imagine two students lifting identical boxes onto a shelf. If both students do the same amount of work, but one student lifts the box faster, that student has greater power.

So:

• More work in the same time means more power.
• The same work in less time also means more power.

8. Calculating Power

To calculate power:

1. Find the work done in joules.
2. Find the time taken in seconds.
3. Divide work by time.

$$P = \frac{W}{t}$$

Worked Example 3: Finding Power

A machine does 600 J of work in 12 s. What is its power?

Step 1: Use the formula.

$$P = \frac{W}{t}$$

Step 2: Substitute the values.

$$P = \frac{600}{12}$$

Step 3: Calculate.

$$P = 50\text{ W}$$

Answer: The machine has a power of 50 W.

Worked Example 4: Work and Power Together

A student lifts a 40 N backpack straight up by 2 m in 4 s. Find:

• the work done
• the power used

Step 1: Find the work.

Because the force and motion are both upward:

$$W = F \times d$$

$$W = 40 \times 2 = 80\text{ J}$$

Step 2: Find the power.

$$P = \frac{W}{t}$$

$$P = \frac{80}{4} = 20\text{ W}$$

Answer:

• Work done = 80 J
• Power used = 20 W

9. Important Ideas to Remember

• Work requires force and movement.
• If there is no movement, no mechanical work is done.
• Work depends on the distance moved in the direction of the force.
• Work is measured in joules (J).
• Power tells how fast work is done.
• Power is measured in watts (W).

10. Common Mistakes

Students often make these mistakes:

• Thinking any force means work is done. Remember: the object must move.
• Using the wrong distance. Use the distance in the direction of the force.
• Mixing up work and power. Work is the amount of energy transferred; power is how fast it happens.
• Forgetting units. Work uses joules, and power uses watts.

11. Quick Check Questions

Try these on your own:

1. A 10 N force moves a ball 3 m in the same direction. How much work is done?
2. A machine does 200 J of work in 5 s. What is its power?
3. You carry a bag across a room at constant height. Is the upward force doing work on the bag in the direction of motion?

Answers:

1. $$W = 10 \times 3 = 30\text{ J}$$
2. $$P = \frac{200}{5} = 40\text{ W}$$
3. No. The upward force is not in the same direction as the horizontal motion, so that force does zero work in the direction of motion.

Summary

Mechanical work happens when a force causes an object to move in the direction of the force. It is calculated using $$W = F \times d$$ and measured in joules.

Power tells how quickly work is done. It is calculated using $$P = \frac{W}{t}$$ and measured in watts.

If you remember that work is energy transferred by force over distance and power is how fast that transfer happens, you will understand the main idea of this topic.

Introduction

When objects move, they have energy because of that motion. This energy is called kinetic energy. A rolling ball, a running athlete, and a moving car all have kinetic energy.

Another important idea is work. In science, work happens when a force causes an object to move in the direction of the force. Work transfers energy to or from an object.

The big idea of this lesson is the Work-Energy Theorem. It says that the net work done on an object equals the change in its kinetic energy.

In symbols, this is written as:

$$W_{\text{net}} = \Delta KE$$

This means that if the total work on an object is positive, its kinetic energy increases and it speeds up. If the total work is negative, its kinetic energy decreases and it slows down.

1. What is kinetic energy?

Kinetic energy is the energy an object has because it is moving. The faster an object moves, the more kinetic energy it has. Also, the more mass an object has, the more kinetic energy it has at the same speed.

The formula for kinetic energy is:

$$KE = \frac{1}{2}mv^2$$

In this formula:

• (m) is mass in kilograms, written as \(kg\)
• (v) is speed in meters per second, written as \(m/s\)
• (KE) is kinetic energy in joules, written as \(J\)

A joule is the unit of energy.

Notice that speed is squared in the formula. That means speed has a very strong effect on kinetic energy. If speed doubles, kinetic energy becomes four times as large.

2. What is work?

Work is done when a force moves an object through a distance. If you push a box and it moves forward, you do work on the box.

The basic formula for work is:

$$W = Fd$$

In this formula:

• (F) is force in newtons, \(N\)
• (d) is distance in meters, \(m\)
• (W) is work in joules, \(J\)

This simple form works when the force and motion are in the same direction.

If the force helps the motion, the work is positive. If the force opposes the motion, the work is negative.

For example:

• A person pushing a sled forward does positive work.
• Friction acting against the moving sled does negative work.

3. Net work

Often, more than one force acts on an object at the same time. The net work is the total work done by all the forces combined.

For example, if one force does \(50\,J\) of work and friction does \(-20\,J\) of work, then the net work is:

$$W_{\text{net}} = 50 + (-20) = 30\,J$$

A positive net work means the object gains kinetic energy. A negative net work means the object loses kinetic energy.

4. The Work-Energy Theorem

The Work-Energy Theorem connects force, motion, and energy in one statement:

$$W_{\text{net}} = \Delta KE = KE_f - KE_i$$

Here:

• (KE_i) is the initial kinetic energy
• (KE_f) is the final kinetic energy
• (\Delta KE) means change in kinetic energy

So the theorem can also be written as:

$$W_{\text{net}} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

This tells us that when net work is done on an object, its speed changes.

5. What positive, negative, and zero net work mean

• Positive net work: The object speeds up, so kinetic energy increases.
• Negative net work: The object slows down, so kinetic energy decreases.
• Zero net work: The kinetic energy stays the same, so the speed does not change.

This does not always mean the object is not moving. Zero net work can mean the object keeps moving at a constant speed.

6. Important relationships to remember

• Kinetic energy depends on mass and speed.
• Work transfers energy.
• Net work changes kinetic energy.
• The unit for both work and energy is the joule.

7. Worked Example 1: Finding kinetic energy

A \(4\,kg\) ball rolls at \(3\,m/s\). Find its kinetic energy.

Step 1: Write the formula.

$$KE = \frac{1}{2}mv^2$$

Step 2: Substitute the values.

$$KE = \frac{1}{2}(4)(3^2)$$

Step 3: Calculate.

$$KE = 2 \cdot 9 = 18\,J$$

Answer: The ball has \(18\,J\) of kinetic energy.

8. Worked Example 2: Using work to find the change in kinetic energy

A student pushes a cart, and the net work done on it is \(60\,J\). If the cart started with \(15\,J\) of kinetic energy, what is its final kinetic energy?

Step 1: Use the Work-Energy Theorem.

$$W_{\text{net}} = KE_f - KE_i$$

Step 2: Substitute known values.

$$60 = KE_f - 15$$

Step 3: Solve for \(KE_f\).

$$KE_f = 75\,J$$

Answer: The cart’s final kinetic energy is \(75\,J\).

9. Worked Example 3: Finding final speed from net work

A \(2\,kg\) object is moving at \(4\,m/s\). The net work done on it is \(20\,J\). Find its final speed.

Step 1: Find the initial kinetic energy.

$$KE_i = \frac{1}{2}mv_i^2 = \frac{1}{2}(2)(4^2) = 16\,J$$

Step 2: Use the Work-Energy Theorem.

$$W_{\text{net}} = KE_f - KE_i$$

$$20 = KE_f - 16$$

$$KE_f = 36\,J$$

Step 3: Use the kinetic energy formula to find final speed.

$$KE_f = \frac{1}{2}mv_f^2$$

$$36 = \frac{1}{2}(2)v_f^2$$

$$36 = v_f^2$$

$$v_f = 6\,m/s$$

Answer: The final speed is \(6\,m/s\).

10. Worked Example 4: Negative work and slowing down

A \(5\,kg\) skateboarder is moving at \(8\,m/s\). Friction does \(-60\,J\) of net work on the skateboarder. What is the skateboarder’s final kinetic energy?

Step 1: Find the initial kinetic energy.

$$KE_i = \frac{1}{2}mv_i^2 = \frac{1}{2}(5)(8^2)$$

$$KE_i = 2.5 \cdot 64 = 160\,J$$

Step 2: Use the Work-Energy Theorem.

$$W_{\text{net}} = KE_f - KE_i$$

$$-60 = KE_f - 160$$

$$KE_f = 100\,J$$

Answer: The final kinetic energy is \(100\,J\). Because the net work is negative, the skateboarder slows down.

11. Common mistakes to avoid

• Forgetting to square the speed in the kinetic energy formula.
• Mixing up mass and weight. In kinetic energy, use mass in kilograms.
• Ignoring the sign of work. Negative work means energy is removed from kinetic energy.
• Using the force from only one source when the problem asks for net work.

12. How to solve Work-Energy problems

1. Identify what is given: mass, speed, force, distance, or work.
2. Find the net work if needed.
3. Use $$W_{\text{net}} = \Delta KE$$.
4. If necessary, use $$KE = \frac{1}{2}mv^2$$ to find a missing speed or energy.
5. Check whether your answer makes sense. Positive net work should increase speed. Negative net work should decrease speed.

13. Why this idea matters

The Work-Energy Theorem helps explain many everyday events. When you kick a soccer ball, your foot does work on the ball and increases its kinetic energy. When brakes stop a bicycle, friction does negative work and removes kinetic energy.

This idea is useful because it lets us study motion using energy, not just forces and acceleration. It gives another powerful way to understand how objects speed up and slow down.

Brief Summary

Kinetic energy is the energy of motion, and it is found using $$KE = \frac{1}{2}mv^2$$. Work is the transfer of energy by a force acting through a distance, and net work is the total work done by all forces.

The Work-Energy Theorem says:

$$W_{\text{net}} = \Delta KE$$

If net work is positive, kinetic energy increases. If net work is negative, kinetic energy decreases. This theorem helps us connect forces, motion, and energy in a clear and useful way.

Gravitational and Elastic Potential Energy

Energy is the ability to cause change or do work. Sometimes energy is stored in an object because of its position or its shape. This stored energy is called potential energy.

In this lesson, you will learn about two important kinds of potential energy:

• Gravitational potential energy — energy stored because an object is above the ground.
• Elastic potential energy — energy stored when something is stretched or compressed, like a spring or rubber band.

These ideas help explain how roller coasters move, why a book on a shelf can fall, and how a stretched bow or spring can launch an object.

1. What is gravitational potential energy?

Gravitational potential energy is the energy an object has because of its height above a reference point, usually the ground. The higher the object is, the more gravitational potential energy it has.

An object also has more gravitational potential energy if it has more mass. A heavy backpack on a shelf stores more gravitational potential energy than a pencil on the same shelf.

The formula for gravitational potential energy is:

where:

• \(GPE\) = gravitational potential energy, measured in joules (J)
• \(m\) = mass, measured in kilograms (kg)
• \(g\) = gravitational field strength, usually \(9.8\,\text{N/kg}\) on Earth (often rounded to \(10\,\text{N/kg}\) for simple calculations)
• \(h\) = height, measured in meters (m)

This means gravitational potential energy depends on mass, gravity, and height.

2. Understanding the formula \(GPE = mgh\)

If any of the values in the formula increase, the gravitational potential energy increases too.

• If mass increases, \(GPE\) increases.
• If height increases, \(GPE\) increases.
• If gravity is stronger, \(GPE\) increases.

On Earth, \(g\) usually stays the same, so most school problems focus on how mass and height affect the energy.

3. What is elastic potential energy?

Elastic potential energy is the energy stored when an object is stretched or compressed. This happens in springs, rubber bands, trampolines, and even some toys.

When you stretch a spring, you do work on it. That work is stored as elastic potential energy. When the spring is released, the stored energy can turn into motion.

For a spring, the formula is:

where:

• \(EPE\) = elastic potential energy, measured in joules (J)
• \(k\) = spring constant, measured in newtons per meter (N/m)
• \(x\) = distance stretched or compressed from its normal length, measured in meters (m)

The spring constant \(k\) tells how stiff the spring is. A larger \(k\) means the spring is harder to stretch or compress.

4. Important ideas about elastic potential energy

• If a spring is stretched more, it stores more energy.
• If the spring is stiffer, it stores more energy for the same stretch distance.
• The distance \(x\) is squared, so doubling the stretch does not just double the energy — it makes the energy four times as large.

For example, if \(x\) changes from \(0.10\,\text{m}\) to \(0.20\,\text{m}\), the elastic potential energy becomes four times greater, not two times greater.

5. Units of energy

Both gravitational potential energy and elastic potential energy are measured in joules, written as \(J\).

One joule is a small amount of energy, so in real-life situations you may see larger numbers. Always make sure your measurements are in the correct units before using the formulas.

• Mass should be in kilograms, not grams.
• Height should be in meters, not centimeters.
• Stretch distance should be in meters, not centimeters.

If needed, convert units first:

• \(1000\,\text{g} = 1\,\text{kg}\)
• \(100\,\text{cm} = 1\,\text{m}\)

6. Energy transformations

Potential energy often changes into other forms of energy. A raised object can fall, changing gravitational potential energy into kinetic energy. A stretched spring can snap back, changing elastic potential energy into kinetic energy.

For example:

• A book on a shelf has gravitational potential energy. When it falls, that energy changes into motion.
• A compressed spring in a toy stores elastic potential energy. When released, it pushes the toy forward.

This idea is part of energy conservation: energy is not created or destroyed, but it can change form.

7. Worked Example 1: Finding gravitational potential energy

Question: A \(2\,\text{kg}\) book is placed on a shelf \(1.5\,\text{m}\) high. What is its gravitational potential energy? Use \(g = 9.8\,\text{N/kg}\).

Step 1: Write the formula.

Step 2: Substitute the values.

Step 3: Calculate.

Answer: The book has \(29.4\,\text{J}\) of gravitational potential energy.

8. Worked Example 2: Comparing heights

Question: A \(5\,\text{kg}\) object is lifted from \(2\,\text{m}\) to \(6\,\text{m}\). How much does its gravitational potential energy increase? Use \(g = 10\,\text{N/kg}\).

Step 1: Find the change in height.

Step 2: Use the increase in height in the formula.

Step 3: Substitute the values.

Step 4: Calculate.

Answer: The object's gravitational potential energy increases by \(200\,\text{J}\).

9. Worked Example 3: Finding elastic potential energy

Question: A spring has a spring constant of \(200\,\text{N/m}\). It is compressed by \(0.10\,\text{m}\). How much elastic potential energy is stored?

Step 1: Write the formula.

Step 2: Substitute the values.

Step 3: Square the distance.

Step 4: Calculate.

Answer: The spring stores \(1\,\text{J}\) of elastic potential energy.

10. Worked Example 4: A more challenging spring problem

Question: A spring with \(k = 150\,\text{N/m}\) is stretched by \(0.20\,\text{m}\). Find the elastic potential energy.

Step 1: Use the formula.

Step 2: Substitute.

Step 3: Square the stretch distance.

Step 4: Multiply.

Answer: The spring stores \(3\,\text{J}\) of elastic potential energy.

11. Common mistakes to avoid

• Using the wrong units: Always convert grams to kilograms and centimeters to meters.
• Forgetting to square \(x\): In elastic potential energy, the stretch or compression distance must be squared.
• Using total height incorrectly: Sometimes you need the change in height, not just one height value.
• Leaving out \(g\): In gravitational potential energy, you must include gravity.

12. Quick comparison

13. Practice thinking

Ask yourself these questions when solving a problem:

1. What kind of potential energy is involved?
2. Do I know the correct formula?
3. Are my units in kilograms and meters?
4. Am I solving for total energy or change in energy?

14. Summary

Potential energy is stored energy. Gravitational potential energy depends on mass, gravity, and height, and is found using \(GPE = mgh\).

Elastic potential energy is stored in stretched or compressed objects like springs, and is found using $$EPE = \frac{1}{2}kx^2$$.

Both types of energy are measured in joules, and both can change into other forms of energy such as motion. If you remember the formulas, use correct units, and identify what is changing, you can solve these problems successfully.

Global Energy Infrastructure is a big name for something very important: the systems people use all around the world to make, move, and use energy.

Energy helps us do many things every day. It powers lights, refrigerators, schools, buses, computers, and hospitals. But energy does not usually start in our homes. It travels through a chain of places and machines before it gets to us.

In this lesson, you will learn how energy moves from its source to people, what kinds of energy sources we use, and how different sources can help or harm Earth.

Introduction: What is energy infrastructure?

Infrastructure means the important systems that help a community work. Roads are part of transportation infrastructure. Pipes are part of water infrastructure. Wires, power plants, and fuel delivery systems are part of energy infrastructure.

Global energy infrastructure means these systems are used all over the world. Different countries use different energy sources depending on what they have nearby, what technology they can build, and what choices they make.

Some places use lots of coal or oil. Some places use water, wind, or sunlight. Many places use a mix of energy sources.

Main Teaching Point 1: The energy supply chain

An energy supply chain is the path energy takes from where it begins to where people use it.

Here is a simple energy supply chain for electricity:

1. Energy source is found or collected.
2. The source is used in a power plant or machine.
3. Electricity is made.
4. Electricity travels through power lines.
5. Homes, schools, and businesses use the electricity.

Not all energy supply chains are the same. Gasoline for cars has a different path:

1. Oil is taken from deep underground or under the ocean.
2. The oil is moved to a refinery.
3. The refinery changes the oil into fuels like gasoline.
4. Trucks or pipes carry the fuel to gas stations.
5. Cars use the fuel.

So, global energy infrastructure includes things like:

• power plants
• solar panels
• wind turbines
• dams
• oil wells
• pipelines
• power lines
• batteries
• gas stations

Main Teaching Point 2: Main energy sources

People use several main kinds of energy sources. Let’s look at the biggest groups.

Fossil fuels include coal, oil, and natural gas. These fuels formed from living things long ago, over a very long time. People burn them to make electricity, heat, and fuel for transportation.

Nuclear energy comes from a special fuel, often uranium, inside a power plant. The plant uses heat from the fuel to make steam, and the steam helps make electricity.

Renewable energy comes from sources that can be used again and again, such as sunlight, wind, and moving water.

Each energy source has strengths and limits.

Main Teaching Point 3: Fossil fuel systems

Fossil fuels are used in many countries because they can make a lot of energy.

Common fossil fuels are:

• Coal — burned in some power plants
• Oil — turned into fuels like gasoline and diesel
• Natural gas — used for electricity, heating, and cooking

How fossil fuel infrastructure works:

1. Fuel is taken from Earth.
2. It is moved by train, truck, ship, or pipeline.
3. It is burned in a power plant, factory, home, or vehicle.
4. The energy is used to make electricity, heat, or motion.

Good points:

• They can provide large amounts of energy.
• They have been used for a long time, so many systems already exist.
• Energy can often be made whenever people need it.

Problems:

• Burning them can make air pollution.
• They add gases to the air that warm Earth.
• They are not renewable, so they can run out.
• Getting them from Earth can harm land and water.

Main Teaching Point 4: Nuclear energy systems

Nuclear power plants make electricity in a different way. They use fuel that releases heat. That heat turns water into steam. The steam spins a turbine, and the turbine helps make electricity.

How nuclear infrastructure works:

1. Nuclear fuel is prepared.
2. The fuel is used in a power plant.
3. Heat makes steam.
4. Steam spins a turbine.
5. Electricity travels through power lines.

Good points:

• It can make a lot of electricity.
• It does not burn coal, oil, or gas to make electricity.
• It can run for long periods of time.

Problems:

• Nuclear plants are expensive and take a long time to build.
• The waste must be stored carefully.
• Plants need very strong safety systems.

Nuclear energy is used in some parts of the world, but not everywhere.

Main Teaching Point 5: Renewable energy systems

Renewable energy comes from sources that nature keeps providing.

Solar energy uses sunlight. Solar panels collect light energy and change it into electricity.

Wind energy uses moving air. Wind turns the blades of a turbine, and the turbine helps make electricity.

Hydropower uses moving water. Water flows through a dam or river system and spins turbines.

Good points:

• These sources can be used again and again.
• They usually make less air pollution while producing electricity.
• They can help reduce harm to Earth.

Problems:

• Solar panels need sunlight, so they work best on sunny days.
• Wind turbines need wind.
• Hydropower needs enough moving water.
• Some renewable systems need lots of space.

Because the weather changes, renewable energy does not always make the same amount of electricity every hour. This is one reason batteries and backup systems are important.

Main Teaching Point 6: Energy grids

An energy grid is a network that moves electricity from where it is made to where it is used.

The grid includes:

• power plants
• wires and towers
• stations that help control electricity
• homes, schools, and buildings that use the electricity

Think of the grid like a giant road system, but for electricity instead of cars.

In many places, the grid connects different energy sources together. A city might get electricity from a gas plant, a wind farm, and a solar field all on the same day.

This is helpful because if one source is making less energy, another source may help.

Main Teaching Point 7: Efficiency

Efficiency means how well something does its job without wasting too much energy.

If two systems both try to provide electricity, but one loses less energy along the way, that system is more efficient.

For example, energy can be lost as heat when electricity moves through wires or when fuel is burned in machines.

We can think of efficiency in a simple way like this:

$$\text{Efficiency} = \frac{\text{useful energy out}}{\text{energy in}}$$

You do not need to memorize the formula, but it helps show that we want as much useful energy as possible.

A more efficient system:

• wastes less energy
• can save money
• can reduce pollution

Main Teaching Point 8: Environmental impact

Environmental impact means how something affects nature, including air, water, land, plants, animals, and people.

Different energy systems affect the environment in different ways.

• Fossil fuels can cause more air pollution and add gases that warm Earth.
• Nuclear makes electricity without burning fossil fuels, but its waste must be handled safely.
• Renewables often cause less air pollution while making electricity, but building dams, wind farms, or large solar fields can still change habitats.

No energy source is perfect. That is why many countries try to choose a mix that gives enough energy and also protects the environment.

Main Teaching Point 9: Technological limits

Technological limits are the things technology cannot do easily yet.

For example:

• Batteries cannot always store enough energy for a whole city for a long time.
• Some power lines are old and need repair.
• Some places cannot afford to build new power plants quickly.
• Some renewable sources depend on weather and location.

If a place has little wind, wind turbines may not help much there. If a place gets lots of sun, solar power may work very well.

This is why energy choices are different in different parts of the world.

Main Teaching Point 10: Why countries use a mix of energy sources

Many countries do not rely on only one source of energy. They use a mix.

A mix can help because:

• it gives more reliable energy
• it can lower pollution
• it can make use of local resources
• it can provide backup if one source is unavailable

For example, a country might use:

• solar during sunny daytime hours
• wind when it is windy
• hydropower from rivers
• natural gas or nuclear power when steady electricity is needed

Using a mix helps the grid stay strong.

Worked Example 1: Following an energy path

Question: A lamp in a house turns on using electricity from a wind farm. What is the path the energy takes?

Step 1: Start with the source. The source is moving air, or wind.

Step 2: Wind turns the blades of a turbine.

Step 3: The turbine helps make electricity.

Step 4: The electricity travels through power lines on the grid.

Step 5: The house receives the electricity, and the lamp lights up.

Answer: Wind → turbine → electricity → power lines → house → lamp.

Worked Example 2: Comparing two energy sources

Question: Which source is renewable: coal or sunlight?

Step 1: Ask if the source can be replaced naturally in a short time.

Step 2: Coal takes a very long time to form, so it is not renewable.

Step 3: Sunlight keeps coming each day, so it is renewable.

Answer: Sunlight is renewable. Coal is not.

Worked Example 3: Thinking about environmental impact

Question: A town wants cleaner air. Should it think about adding more solar power or burning more coal?

Step 1: Compare how each source affects the air while making electricity.

Step 2: Burning coal can make more air pollution.

Step 3: Solar panels make electricity from sunlight and usually cause less air pollution while operating.

Answer: The town should think about adding more solar power if it wants cleaner air.

Worked Example 4: Choosing an energy mix

Question: A place is sunny in the day, windy at night, and has a river nearby. Why might using more than one energy source be helpful?

Step 1: In the day, solar power can work well.

Step 2: At night, solar power does not make electricity, but wind may help.

Step 3: The river can provide hydropower as another source.

Step 4: Using all three can make the energy supply more steady.

Answer: A mix of solar, wind, and hydropower can help provide electricity at different times and make the grid stronger.

Key ideas to remember

• Global energy infrastructure is the worldwide system that makes, moves, and delivers energy.
• An energy supply chain shows how energy travels from source to user.
• Fossil fuels, nuclear energy, and renewable energy are major ways people make electricity.
• The energy grid carries electricity to homes, schools, and businesses.
• Efficiency means wasting less energy.
• Environmental impact means how energy choices affect Earth.
• Different energy sources have different strengths and limits.
• Many places use a mix of energy sources to keep energy reliable.

Brief Summary

Energy does not appear by magic in our homes. It comes through a global system of sources, machines, wires, fuel paths, and workers. People around the world use fossil fuels, nuclear energy, and renewable energy in different ways.

Each source can help provide the energy people need, but each also has limits. By understanding supply chains, grids, efficiency, and environmental impact, we can better understand how the world powers daily life.

Conservation of Momentum helps us understand what happens when objects collide or push away from each other. Even though the motion of each object may change, the total momentum of the whole system stays the same if no outside force changes it.

This idea is very useful in science because it helps explain car crashes, sports hits, billiard balls, skating, and even rockets in space. In this lesson, you will learn what momentum is, what it means to be conserved, and how to use simple math to solve collision problems.

What is momentum?

Momentum is the amount of motion an object has. An object with more mass or more speed has more momentum.

The formula for momentum is:

$$p = m \times v$$

where:

• p = momentum
• m = mass
• v = velocity

Velocity means speed with direction. Direction matters for momentum. For example, moving to the right can be positive, and moving to the left can be negative.

This means two objects can have the same mass and speed, but if they move in opposite directions, their momentum values will be different because one may be positive and the other negative.

Why direction matters

Suppose a ball has momentum of \(+10\) and another ball has momentum of \(-10\). Together, their total momentum is:

$$+10 + (-10) = 0$$

So momentum is not just about how much motion there is. It also includes which way the object is moving.

What does conservation mean?

To conserve something means to keep the total amount the same. The law of conservation of momentum says:

$$\text{Total momentum before} = \text{Total momentum after}$$

This rule works in a closed system. A closed system means no outside force changes the total momentum in an important way during the event.

For many 7th grade problems, we treat the colliding objects as a closed system while they collide.

The conservation of momentum equation

For two objects, we can write:

$$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$$

Here:

• \(m_1\) and \(m_2\) are the masses of the two objects
• \(v_1\) and \(v_2\) are their velocities before the collision
• \(v_1'\) and \(v_2'\) are their velocities after the collision

The left side shows total momentum before the collision. The right side shows total momentum after the collision.

Closed systems and outside forces

If outside forces are very small or act for only a very short time, we can often ignore them. Then the system acts like a closed system, and momentum is conserved.

For example, when two skateboards push off each other, we usually focus on the two riders as the system. Their total momentum stays the same as long as outside forces do not strongly affect the motion during the push.

Elastic and inelastic collisions

There are different kinds of collisions, but momentum is conserved in both of these:

• Elastic collision: The objects bounce apart after colliding.
• Inelastic collision: The objects do not bounce apart normally. They may stick together or move off with less bounce.

The important idea for this lesson is that the total momentum stays the same in both types, as long as the system is closed.

In an elastic collision, objects bounce off each other. A cue ball hitting another billiard ball is a common example.

In an inelastic collision, the objects may stick together. A lump of clay hitting another lump of clay and sticking is a simple example.

How to solve momentum problems

1. Find the momentum of each object before the collision using \(p = mv\).
2. Add the momenta carefully, using positive and negative signs for direction.
3. Set total momentum before equal to total momentum after.
4. Solve for the missing velocity or momentum.
5. Check whether your answer makes sense.

Worked Example 1: Finding momentum

A cart with mass \(2\,\text{kg}\) moves to the right at \(3\,\text{m/s}\). What is its momentum?

Step 1: Use the formula

$$p = mv$$

Step 2: Substitute the values

$$p = 2 \times 3 = 6$$

Answer: The momentum is \(6\,\text{kg}\cdot\text{m/s}\) to the right.

If we choose right as positive, we can write the momentum as \(+6\,\text{kg}\cdot\text{m/s}\).

Worked Example 2: Two objects stick together

A \(2\,\text{kg}\) cart moving right at \(4\,\text{m/s}\) crashes into a \(1\,\text{kg}\) cart at rest. The carts stick together. What is their final velocity?

Step 1: Find total momentum before the collision

First cart momentum:

$$p_1 = 2 \times 4 = 8$$

Second cart momentum:

$$p_2 = 1 \times 0 = 0$$

Total momentum before:

$$8 + 0 = 8$$

Step 2: Use conservation of momentum

After the collision, the carts stick together, so their total mass is:

$$2 + 1 = 3\,\text{kg}$$

Let the final velocity be \(v\).

$$3v = 8$$

Step 3: Solve

$$v = \frac{8}{3} \approx 2.7\,\text{m/s}$$

Answer: The stuck-together carts move at about \(2.7\,\text{m/s}\) to the right.

This is an inelastic collision because the carts stick together.

Worked Example 3: Objects moving in opposite directions

A \(3\,\text{kg}\) cart moves right at \(2\,\text{m/s}\). A \(1\,\text{kg}\) cart moves left at \(1\,\text{m/s}\). They stick together. What is their final velocity?

Step 1: Choose directions

Let right be positive and left be negative.

Step 2: Find each momentum

First cart:

$$p_1 = 3 \times 2 = +6$$

Second cart:

$$p_2 = 1 \times (-1) = -1$$

Total momentum before:

$$+6 + (-1) = +5$$

Step 3: Find total mass after collision

$$3 + 1 = 4\,\text{kg}$$

Step 4: Solve for final velocity

$$4v = 5$$

$$v = \frac{5}{4} = 1.25\,\text{m/s}$$

Answer: The carts move together at \(1.25\,\text{m/s}\) to the right.

Even though one cart was moving left, the total momentum was still positive, so the final motion is to the right.

Worked Example 4: Elastic collision idea

A \(1\,\text{kg}\) ball moves right at \(5\,\text{m/s}\). It collides with a \(1\,\text{kg}\) ball at rest. After the collision, the first ball stops. What is the velocity of the second ball?

Step 1: Find total momentum before

First ball:

$$1 \times 5 = 5$$

Second ball:

$$1 \times 0 = 0$$

Total momentum before:

$$5 + 0 = 5$$

Step 2: Find total momentum after

After the collision, the first ball stops, so its momentum is \(0\).

Let the second ball's velocity be \(v\).

$$1 \times v = 5$$

So:

$$v = 5\,\text{m/s}$$

Answer: The second ball moves right at \(5\,\text{m/s}\).

This is a simple example of an elastic collision, where the objects bounce apart instead of sticking.

Important ideas to remember

• Momentum depends on mass and velocity.
• The formula is \(p = mv\).
• Velocity includes direction, so momentum can be positive or negative.
• In a closed system, total momentum before equals total momentum after.
• Momentum is conserved in both elastic and inelastic collisions.
• If objects stick together, add their masses for the final motion.

Common mistakes

• Forgetting direction: Left and right should not be treated the same. Use positive and negative signs.
• Mixing up mass and momentum: Mass is how much matter is in an object. Momentum is mass times velocity.
• Forgetting the total system: Conservation applies to the total momentum of all objects together.
• Not using the final combined mass: If two objects stick together, they move as one object after the collision.

Real-world connections

• Skaters pushing apart: One skater moves backward while the other moves forward, but the total momentum stays balanced.
• Car collisions: Investigators can study momentum to understand what happened during a crash.
• Sports: A bat hitting a ball or a player catching a pass shows momentum changing between objects.
• Rockets: Gas is pushed backward, and the rocket moves forward.

Brief Summary

Momentum is the amount of motion an object has, and it is found using \(p = mv\). Because velocity includes direction, momentum also has direction.

In a closed system, the total momentum stays the same before and after a collision. This is called the conservation of momentum.

Whether objects bounce apart in an elastic collision or stick together in an inelastic collision, the total momentum is still conserved. If you keep track of mass, velocity, and direction, you can solve many collision problems.

Conservation of Mechanical Energy explains how energy can change form while the total mechanical energy stays the same in a system with no friction or air resistance.

In many moving systems, energy shifts back and forth between potential energy and kinetic energy. A roller coaster at the top of a hill has a lot of potential energy. As it moves downward, that potential energy changes into kinetic energy, which is the energy of motion.

This idea helps us predict how fast objects move and how high they can rise without needing to track every force step by step. If no energy is lost to friction, then the total amount of mechanical energy remains constant.

1. What is mechanical energy?

Mechanical energy is the total of an object's kinetic energy and potential energy.

We can write this as:

$$ME = KE + PE$$

Where:

• (Kinetic energy) is energy due to motion
• (Potential energy) is stored energy due to position

For most 9th Grade problems, the two main equations are:

$$KE = \frac{1}{2}mv^2$$

$$PE = mgh$$

•  is mass in kilograms (kg)
•  is speed in meters per second (m/s)
•  is gravitational field strength, usually  on Earth
•  is height in meters (m)

On Earth, we often use:

$$g = 9.8\,m/s^2$$

Sometimes in simple school problems,  may be rounded to .

2. What does conservation mean?

Conservation means the total amount stays the same. In a frictionless system:

$$KE_{initial} + PE_{initial} = KE_{final} + PE_{final}$$

This is the law of conservation of mechanical energy.

If an object loses potential energy, it gains the same amount of kinetic energy. If it gains potential energy, it loses kinetic energy.

3. When can we use conservation of mechanical energy?

We use it when:

• Friction is absent or so small that we ignore it
• Air resistance is absent or ignored
• We are looking at energy changing between motion and position

Common examples include:

• A falling object
• A ball thrown upward
• A pendulum swinging
• A roller coaster moving along a track

If friction is present, some mechanical energy changes into heat and sound. Then mechanical energy is not conserved, even though total energy overall is still conserved.

4. How energy changes in motion

Imagine a skateboarder at the top of a ramp. At the top:

• Potential energy is high
• Kinetic energy is low if the skateboarder starts from rest

As the skateboarder moves down:

• Height decreases, so potential energy decreases
• Speed increases, so kinetic energy increases

At the bottom:

• Potential energy is lowest
• Kinetic energy is highest

If the skateboarder then goes up another ramp, kinetic energy changes back into potential energy.

5. A helpful way to think about it

Mechanical energy is like money being moved between two accounts. One account is kinetic energy, and the other is potential energy. In a frictionless system, the total amount of money stays the same. It just shifts from one account to the other.

6. Important ideas to remember

• Mass affects both kinetic and potential energy.
• Height affects gravitational potential energy.
• Speed affects kinetic energy, and because speed is squared, small speed changes can make a big difference.
• The lowest point in motion usually has the greatest kinetic energy.
• The highest point usually has the greatest potential energy.

7. Solving conservation of mechanical energy problems

Use these steps:

1. Choose the starting point and ending point.
2. Write the energy equation: $$KE_i + PE_i = KE_f + PE_f$$
3. Substitute the formulas for kinetic and potential energy.
4. Fill in the known values.
5. Solve for the unknown quantity, such as speed or height.

Worked Example 1: A falling rock

A rock of mass  is dropped from a height of . Ignore air resistance. What is its speed just before it hits the ground?

Step 1: Identify the energies.

At the top, the rock starts from rest, so:

• 
• 

At the ground:

•  because height is zero
•  is what we want

Step 2: Use conservation of mechanical energy.

$$KE_i + PE_i = KE_f + PE_f$$

$$0 + mgh = \frac{1}{2}mv^2 + 0$$

Substitute the values:

$$2(9.8)(5) = \frac{1}{2}(2)v^2$$

$$98 = v^2$$

$$v = \sqrt{98} \approx 9.9\,m/s$$

Answer: The rock hits the ground at about 9.9 m/s.

Notice: The mass cancels out. In a frictionless fall, the speed depends on height, not mass.

Worked Example 2: A ball thrown upward

A ball is thrown straight up at . How high does it go before stopping for an instant? Ignore air resistance.

Step 1: Think about the top of the motion.

At the highest point, the ball's speed is zero, so all of its kinetic energy has changed into potential energy.

Step 2: Use conservation of mechanical energy.

$$\frac{1}{2}mv^2 + 0 = 0 + mgh$$

Substitute  and :

$$\frac{1}{2}m(14)^2 = m(9.8)h$$

$$98m = 9.8mh$$

Divide both sides by :

$$98 = 9.8h$$

$$h = 10\,m$$

Answer: The ball rises to a maximum height of 10 m.

Worked Example 3: Roller coaster on a hill

A roller coaster car starts from rest at the top of a  hill. What is its speed when it reaches a point that is  above the ground? Ignore friction.

Step 1: Write the starting and ending energies.

Initial height: 

Final height: 

Initial speed: 

So:

$$0 + mg(30) = \frac{1}{2}mv^2 + mg(10)$$

Step 2: Simplify.

$$mg(20) = \frac{1}{2}mv^2$$

The mass cancels:

$$g(20) = \frac{1}{2}v^2$$

$$2(9.8)(20) = v^2$$

$$392 = v^2$$

$$v \approx 19.8\,m/s$$

Answer: The coaster's speed is about 19.8 m/s.

Worked Example 4: Pendulum energy

A pendulum bob is released from rest. At its highest point, it is  above its lowest point. What is its speed at the lowest point? Ignore air resistance.

Step 1: Set up the energy change.

At the top:

• 
• 

At the bottom:

• 
• 

Use:

$$mgh = \frac{1}{2}mv^2$$

Substitute  and :

$$m(9.8)(2.0) = \frac{1}{2}mv^2$$

$$19.6 = \frac{1}{2}v^2$$

$$39.2 = v^2$$

$$v \approx 6.3\,m/s$$

Answer: The pendulum bob moves at about 6.3 m/s at the lowest point.

8. Common mistakes to avoid

• Forgetting that speed is squared in kinetic energy.
• Using the wrong height. Height should be measured from a chosen reference level.
• Including friction in a problem that says to ignore it, or ignoring friction when it matters.
• Mixing up mass and weight. Use mass in kilograms in the equations.
• Thinking energy disappears. In these problems, energy changes form; it does not vanish.

9. Quick check questions

• If a ball rolls downhill without friction, what happens to its potential energy?
• Where does a pendulum have the most kinetic energy?
• If a coaster climbs higher, what happens to its kinetic energy?
• In a frictionless system, why does total mechanical energy stay constant?

10. Final summary

The conservation of mechanical energy says that in a frictionless system, the total of kinetic energy and potential energy remains the same.

As an object moves, energy can transfer between potential energy and kinetic energy. When height decreases, potential energy decreases and kinetic energy usually increases. When height increases, kinetic energy usually decreases and potential energy increases.

This principle is very useful for understanding falling objects, roller coasters, pendulums, and many other types of motion.

Thermodynamics and Heat Transfer Mechanisms

In science, thermodynamics is the study of heat, temperature, and how energy moves from one place to another. A big idea in thermodynamics is that energy naturally moves from a place with a higher temperature to a place with a lower temperature.

This lesson focuses on thermal equilibrium and the three main ways heat is transferred: conduction, convection, and radiation. These ideas help explain everyday events, like why a metal spoon gets hot in soup, why warm air rises, and how the Sun heats Earth.

Important idea: Heat is the transfer of thermal energy, while temperature tells how hot or cold something is. Temperature measures the average kinetic energy of particles, but heat is energy moving because of a temperature difference.

Thermal Equilibrium

When two objects at different temperatures touch or interact, heat moves from the warmer object to the cooler object. This continues until both objects reach the same temperature. At that point, they are in thermal equilibrium.

For example, if you place a cold can of juice on a kitchen counter, the juice warms up and the air around it cools slightly. After enough time, the can and the room reach about the same temperature. Then there is no longer a net flow of heat between them.

Thermal equilibrium does not mean energy stops existing. It means the energy transfer has balanced out so there is no overall heat flow from one object to the other.

The Direction of Heat Transfer

• Heat always moves from warmer to cooler objects.
• Heat transfer continues until temperatures become equal.
• The total energy is conserved, but it may spread out between objects.

If object A is at \(80^\circ\text{C}\) and object B is at \(20^\circ\text{C}\), heat will flow from A to B. It will not naturally flow from the colder object to the hotter one.

Heat Transfer Mechanism 1: Conduction

Conduction is the transfer of heat through direct contact of particles. It happens most clearly in solids, where particles are packed closely together.

When one part of a solid is heated, its particles vibrate faster. These particles bump into nearby particles and transfer energy to them. In this way, heat moves through the material.

Metals are usually good conductors of heat. That is why a metal pan heats up quickly on a stove. Materials like wood, plastic, and foam are poor conductors, so they are called insulators.

Examples of conduction:

• A metal spoon left in hot soup becomes hot.
• Your hand feels cold when you touch an ice cube because heat moves from your hand to the ice.
• A pan on a stove transfers heat from the bottom of the pan to the handle.

Heat Transfer Mechanism 2: Convection

Convection is the transfer of heat in fluids. Fluids include liquids and gases. In convection, heat is transferred by the movement of the fluid itself.

When a fluid is heated, its particles move faster and spread out. This makes that part of the fluid less dense, so it rises. Cooler, denser fluid sinks down. This movement creates a convection current.

Convection is why air circulates in a room and why water in a pot moves as it boils.

Examples of convection:

• Boiling water circulates as hot water rises and cooler water sinks.
• Warm air rises from a heater and cooler air moves in to replace it.
• Sea breezes form because land and water heat at different rates, causing air movement.

Heat Transfer Mechanism 3: Radiation

Radiation is the transfer of energy by electromagnetic waves. Unlike conduction and convection, radiation does not need matter to travel through. This means radiation can transfer energy through a vacuum.

The most important example is the Sun. Energy from the Sun travels through the empty space of space and reaches Earth by radiation.

Dark, dull surfaces usually absorb radiation better than light, shiny surfaces. That is why black clothing often feels hotter in sunlight.

Examples of radiation:

• Feeling warmth from the Sun on your skin.
• Feeling heat from a campfire without touching it.
• A road becoming hot in sunlight.

Comparing the Three Heat Transfer Mechanisms

• Conduction: heat transfer by direct contact, mainly in solids.
• Convection: heat transfer by movement of fluids, in liquids and gases.
• Radiation: heat transfer by waves, can happen through empty space.

Particle View of Heat Transfer

All matter is made of particles. When an object is heated, its particles gain kinetic energy and move faster. This particle model helps explain all three heat transfer methods.

• In conduction, faster particles collide with nearby particles.
• In convection, warmer particles spread out, making the fluid rise and move.
• In radiation, energy is sent out as waves rather than by particle collisions.

Thermal Conductors and Insulators

A thermal conductor allows heat to move through it easily. A thermal insulator slows down heat transfer.

Examples of conductors: copper, aluminum, iron.

Examples of insulators: wool, plastic, rubber, air, foam.

Insulators are useful when we want to reduce heat transfer. For example, oven mitts protect your hands, and foam cups help keep drinks hot or cold longer.

Heat Transfer and Energy Conservation

When heat moves from one object to another, energy is not lost. It is transferred. This connects thermodynamics with the larger idea of conservation of energy.

For example, if hot tea cools down, its thermal energy is transferred to the cup and the surrounding air. The energy changes location, but the total amount of energy is still conserved.

A Simple Heat Equation

Sometimes we calculate heat energy using the equation

$$Q = mc\Delta T$$

where:

• \(Q\) = heat energy transferred
• \(m\) = mass
• \(c\) = specific heat capacity
• \(\Delta T\) = change in temperature

At this level, the key idea is that a bigger mass or a bigger temperature change means more heat energy is transferred.

Worked Example 1: Identifying the Type of Heat Transfer

Question: A student holds her hands near a heater and feels warmth without touching it. What type of heat transfer is this?

Step 1: Check whether there is direct contact. There is no contact, so it is not conduction.

Step 2: Think about whether energy can travel across space. The warmth reaching her hands is mainly radiation.

Answer: The main type of heat transfer is radiation.

Worked Example 2: Conduction in a Solid

Question: One end of a metal rod is placed in a flame. After a while, the other end becomes hot. Why?

Step 1: The flame heats particles at one end of the rod.

Step 2: These particles vibrate faster and transfer energy to neighboring particles.

Step 3: The energy moves through the rod from the hot end to the cooler end.

Answer: The rod transfers heat by conduction because the particles in the solid pass energy along through direct contact.

Worked Example 3: Convection Current

Question: Why does warm air rise in a room?

Step 1: When air is heated, its particles move faster and spread out.

Step 2: The warm air becomes less dense than the cooler air around it.

Step 3: The less dense warm air rises, while cooler, denser air sinks.

Answer: Warm air rises because heating causes convection. The air becomes less dense and moves upward.

Worked Example 4: Using \(Q = mc\Delta T\)

Question: A sample of water gains heat and its temperature changes by \(5^\circ\text{C}\). If the mass is \(2\,\text{kg}\) and the specific heat capacity is \(4200\,\text{J/kg}^\circ\text{C}\), how much heat is gained?

Step 1: Write the formula.

$$Q = mc\Delta T$$

Step 2: Substitute the values.

$$Q = (2)(4200)(5)$$

Step 3: Multiply.

$$Q = 42000\,\text{J}$$

Answer: The water gains \(42000\,\text{J}\) of heat energy.

Everyday Applications

• Cooking: pans use conduction, boiling water uses convection, and ovens also transfer heat by radiation.
• Weather: convection in the atmosphere helps create winds and clouds.
• Clothing: jackets trap air, which acts as an insulator and slows heat loss.
• Homes: insulation in walls reduces unwanted heat transfer.
• Space: the Sun warms Earth by radiation through the vacuum of space.

Common Mistakes to Avoid

• Do not confuse heat with temperature. Heat is energy transfer; temperature tells how hot something is.
• Do not say conduction happens through empty space. It needs particles in contact.
• Do not say convection happens in solids. It happens in liquids and gases.
• Do not forget that radiation can happen without matter.
• Do not think cold moves into an object. Instead, heat moves out of the warmer object.

Quick Check for Understanding

1. If a hot mug is left on a table and cools down, in which direction does heat flow?
2. Why does a wooden spoon stay cooler than a metal spoon in hot soup?
3. Which heat transfer mechanism explains boiling water moving in a pot?
4. How does the Sun transfer energy to Earth?
5. What does it mean when two objects are in thermal equilibrium?

Answers:

1. Heat flows from the hot mug to the cooler surroundings.
2. Wood is a better insulator and a poorer conductor than metal.
3. Convection.
4. Radiation.
5. They are at the same temperature, so there is no net heat flow between them.

Summary

Thermodynamics explains how heat energy moves and how objects change temperature. Heat always moves from warmer objects to cooler objects until thermal equilibrium is reached.

The three main heat transfer mechanisms are conduction, convection, and radiation. Conduction happens through direct contact, convection happens through movement in fluids, and radiation transfers energy by waves and can happen through a vacuum.

Understanding these processes helps explain many everyday experiences and connects to the larger science idea that energy is transferred and conserved.

Simple machines are tools that make work easier by changing the size or direction of a force. They do not reduce the total amount of work needed, but they can help you apply force in a more useful way.

In this lesson, you will learn how levers, pulleys, and inclined planes work. You will also learn how to calculate mechanical advantage, including both ideal mechanical advantage and actual mechanical advantage.

These ideas connect to forces and energy because a machine trades one thing for another. Usually, if a machine lets you use less force, you must apply that force over a greater distance.

1. What is a simple machine?

A simple machine is a basic device that changes how a force is used. Common simple machines include:

• Levers
• Pulleys
• Inclined planes

For example, a ramp helps move a heavy box upward with less force than lifting it straight up. A crowbar helps lift an object by increasing the effect of your force. A pulley can make it easier to lift a load by changing the direction of force or sharing the load across several rope sections.

2. What is mechanical advantage?

Mechanical advantage tells how much a machine multiplies the input force. In simple words, it compares the force you put in to the force the machine puts out.

If a machine gives a large output force from a smaller input force, it has a mechanical advantage greater than 1.

The formula for actual mechanical advantage is:

This tells what the machine really does in real life, including friction and other energy losses.

The formula for ideal mechanical advantage depends on distance:

This tells what the machine would do in a perfect situation with no friction.

Important idea: In an ideal machine, the force and distance trade off. If you use less force, you usually must move the machine farther.

3. Work and the distance tradeoff

Work is done when a force moves an object over a distance. In science, work is calculated as:

Here, F is force and d is distance.

Simple machines do not create extra energy. In an ideal machine, the input work equals the output work:

This is why a machine can reduce the force you need, but only if you apply that force over a longer distance.

4. Levers

A lever is a rigid bar that turns around a fixed point called a fulcrum.

The main parts of a lever are:

• Fulcrum: the pivot point
• Effort force: the force you apply
• Resistance force: the load being moved

A lever works by balancing turning effects around the fulcrum. The farther your effort force is from the fulcrum, the easier it is to move the load.

For a lever, ideal mechanical advantage can be found by comparing distances from the fulcrum:

If the effort arm is longer than the resistance arm, the lever gives a mechanical advantage greater than 1.

Examples of levers:

• Crowbar
• Seesaw
• Bottle opener

5. Pulleys

A pulley uses a wheel and rope to lift or move loads.

A pulley can do two main things:

• Change the direction of force
• Reduce the amount of input force needed

In simple pulley systems, the ideal mechanical advantage is often equal to the number of rope segments supporting the load.

For example:

• 1 supporting rope segment: \(IMA = 1\)
• 2 supporting rope segments: \(IMA = 2\)
• 4 supporting rope segments: \(IMA = 4\)

If a pulley system has a higher mechanical advantage, you need less force to lift the object. But you must pull more rope.

6. Inclined planes

An inclined plane is a sloped surface, such as a ramp. It makes it easier to raise an object by spreading the work over a longer distance.

For an inclined plane, ideal mechanical advantage is:

If the ramp is long and not very steep, the mechanical advantage is greater. That means less force is needed to push an object up the ramp.

However, the object must travel a longer distance along the ramp than it would if lifted straight up.

7. Actual vs. ideal mechanical advantage

Real machines are not perfect. Some energy is lost due to friction. Because of this, the actual mechanical advantage is usually less than the ideal mechanical advantage.

Compare them like this:

• IMA: what the machine should do in a perfect world
• AMA: what the machine actually does in real life

If friction is present, you must usually apply more input force than the ideal case predicts.

8. Efficiency

Efficiency tells how well a machine converts input work into useful output work. It is often written as a percent.

A machine with 100% efficiency would waste no energy, but real machines always have less than 100% efficiency.

Worked Example 1: Actual mechanical advantage of a lever

A student uses a lever to lift a rock. The input force is 50 N, and the lever lifts the rock with an output force of 150 N. Find the actual mechanical advantage.

Step 1: Write the formula.

Step 2: Substitute the values.

Step 3: Calculate.

Answer: The actual mechanical advantage is 3. The lever multiplies the input force by 3.

Worked Example 2: Ideal mechanical advantage of an inclined plane

A ramp is 6 m long and rises to a height of 2 m. Find the ideal mechanical advantage.

Step 1: Use the inclined plane formula.

Step 2: Substitute the values.

Step 3: Calculate.

Answer: The ideal mechanical advantage is 3. In an ideal case, the ramp reduces the needed force to one-third of the load force.

Worked Example 3: Pulley force and distance tradeoff

A pulley system has 4 supporting rope segments. A load must be lifted 2 m.

Part A: What is the ideal mechanical advantage?

Part B: How far must the rope be pulled in the ideal case?

When the mechanical advantage is 4, the input distance is 4 times the output distance:

Answer: The ideal mechanical advantage is 4, and the rope must be pulled 8 m.

This shows the distance tradeoff clearly: less force, but more rope must be pulled.

Worked Example 4: Finding efficiency

A machine has an ideal mechanical advantage of 5, but its actual mechanical advantage is 4. Find the efficiency.

Step 1: Use the efficiency formula.

Step 2: Substitute the values.

Step 3: Calculate.

Answer: The machine is 80% efficient.

9. Common mistakes to avoid

• Do not confuse force with work. Force is a push or pull. Work depends on both force and distance.
• Do not mix up AMA and IMA. AMA uses forces. IMA uses distances or machine dimensions.
• Do not forget the tradeoff: a machine that reduces force usually increases distance.
• Do not assume real machines are perfect. Friction usually makes AMA smaller than IMA.

10. How to solve mechanical advantage problems

1. Identify the type of simple machine: lever, pulley, or inclined plane.
2. Decide whether the problem asks for AMA, IMA, or efficiency.
3. Choose the correct formula.
4. Substitute the known values carefully.
5. Check if the answer makes sense. If force goes down, distance should usually go up.

11. Big idea connection

Simple machines are all about using forces in smarter ways. They help us lift, move, or push objects by changing the amount of force needed or the direction of that force.

Mechanical advantage helps us measure how useful a machine is. Ideal mechanical advantage shows the best possible case, while actual mechanical advantage shows what happens in the real world.

Summary

Levers, pulleys, and inclined planes are simple machines that make work easier by changing force and distance. Actual mechanical advantage compares output force to input force, while ideal mechanical advantage compares input distance to output distance or uses machine dimensions.

In all simple machines, there is a tradeoff: using less force usually means applying that force over a greater distance. Real machines also lose some energy to friction, so their actual mechanical advantage is usually lower than their ideal mechanical advantage.

Machine Efficiency and Energy Dissipation is about understanding how well a machine uses the energy put into it.

When you use a machine, not all of the input energy becomes useful output energy. Some energy is almost always “lost” to the surroundings, usually as heat, sound, or unwanted motion. This is called energy dissipation.

In this lesson, you will learn what machine efficiency means, how to calculate it, why no real machine is 100% efficient, and how forces such as friction cause energy to spread out into less useful forms.

1. What is a machine?

A machine is any device that helps us do work more easily. Examples include ramps, pulleys, bicycles, car engines, fans, and even scissors.

Machines do not create energy. Instead, they transfer energy from one form to another or from one place to another.

For example:

• A bicycle changes chemical energy from your body into motion.
• A light bulb changes electrical energy into light and heat.
• A motor changes electrical energy into mechanical energy.

2. Input energy and useful output energy

Every machine has an input and an output.

• Input energy is the total energy supplied to the machine.
• Useful output energy is the energy transferred in the form we actually want.

Sometimes we talk about work input and useful work output instead of energy. Since work is a transfer of energy, the idea is the same.

For example, in a fan:

• Input energy: electrical energy
• Useful output energy: kinetic energy of moving air
• Dissipated energy: heat from the motor and sound

3. What is efficiency?

Efficiency tells us how much of the input energy becomes useful output energy.

The formula for efficiency is:

Efficiency can also be written using work:

To express efficiency as a percentage, multiply by 100:

An efficiency of 100% would mean all input energy becomes useful output energy. In real life, this does not happen because some energy is always dissipated.

4. What is energy dissipation?

Energy dissipation means energy spreads out into the surroundings in forms that are less useful for the task we want.

Usually, dissipated energy becomes:

• thermal energy (heat)
• sound energy
• vibrations
• unwanted motion

The energy is not destroyed. It is still conserved. It has simply changed into forms that are harder to use again.

This matches the law of conservation of energy: energy cannot be created or destroyed, only transferred or transformed.

5. Why are real machines not 100% efficient?

Real machines have parts that rub together or interact with the environment. These interactions cause non-useful energy transfers.

The main reason is often friction. Friction is a non-conservative force, which means it changes useful mechanical energy into thermal energy.

Other causes include:

• air resistance
• deformation of materials
• sound production
• electrical resistance in wires or circuits

For example, when you ride a bike, some of your energy moves the bike forward. But some energy is dissipated because:

• the chain rubs on gears
• the tires rub against the road
• air resistance pushes against you
• parts vibrate and make sound

6. Efficiency and energy conservation together

It is important to remember two ideas at the same time:

1. Total energy is conserved.
2. Useful energy is often less than input energy.

That means:

This is a very helpful relationship when solving problems.

If you know any two of these quantities, you can find the third.

7. Worked Example 1: Finding efficiency from energy values

A machine takes in 500 J of energy and gives 400 J of useful output energy. Find its efficiency.

Step 1: Write the formula.

Step 2: Substitute the numbers.

Step 3: Calculate.

Step 4: Convert to a percentage.

Answer: The machine is 80% efficient.

This means 80% of the energy becomes useful output, and 20% is dissipated.

8. Worked Example 2: Finding dissipated energy

An electric motor receives 1200 J of electrical energy. It produces 900 J of useful mechanical energy. How much energy is dissipated?

Use the relationship:

Rearrange to find dissipated energy:

Substitute the values:

Answer: 300 J of energy is dissipated.

This dissipated energy may become heat and sound.

9. Worked Example 3: Finding useful output from efficiency

A machine is 65% efficient and receives 2000 J of input energy. Find the useful output energy.

Step 1: Change the percentage to a decimal.

Step 2: Use the formula.

Rearrange:

Step 3: Substitute the values.

Step 4: Calculate.

Answer: The useful output energy is 1300 J.

The rest is dissipated:

10. Worked Example 4: Efficiency using work

A student uses a simple machine to lift a box. The student does 600 J of work on the machine, and the machine does 450 J of useful work on the box. Find the efficiency.

Use the work version of the formula:

Substitute the values:

Calculate:

Answer: The machine is 75% efficient.

11. How friction causes energy dissipation

Friction is a force that opposes motion between surfaces that touch. When friction acts, some of the energy of motion is transformed into thermal energy.

Suppose you slide a book across a desk. The book slows down because friction acts opposite to its motion. The kinetic energy of the book is not destroyed. Instead, much of it is transferred as heat to the book, desk, and surrounding air.

That is why friction lowers the efficiency of many machines.

12. Non-conservative forces

In 9th Grade science, you can think of non-conservative forces as forces that cause energy to change into forms like heat or sound, making it less useful for mechanical tasks.

Examples include:

• friction
• air resistance

These forces do not remove energy from the universe, but they do reduce the amount of useful mechanical energy in a system.

13. Everyday examples of efficiency and dissipation

• Car engine: Fuel provides input energy. Some becomes useful motion, but much is dissipated as heat and sound.
• Light bulb: Electrical energy goes in. Light is the useful output, but heat is also produced.
• Bicycle: Your body provides input energy. Some becomes forward motion, while some is dissipated through friction and air resistance.
• Phone charger: Electrical energy charges the battery, but some energy is dissipated as heat.

14. How can efficiency be improved?

Machines can be made more efficient by reducing unwanted energy transfers.

Ways to improve efficiency include:

• lubricating moving parts to reduce friction
• using smoother surfaces
• designing shapes that reduce air resistance
• using materials that waste less energy as heat
• maintaining machines so they work properly

Even with improvements, no real machine can be perfectly efficient.

15. Common mistakes to avoid

• Forgetting to multiply by 100 when giving efficiency as a percentage.
• Mixing up input and output. Input is the total energy supplied; useful output is the energy you want.
• Thinking dissipated energy is destroyed. It is not destroyed; it is transferred to less useful forms.
• Assuming 100% efficiency is possible in real life. Real machines always have some energy dissipation.

16. Quick check questions

1. A machine takes in 800 J and gives 600 J of useful energy. What is its efficiency?
2. If a machine has 1000 J input and 250 J is dissipated, how much useful output energy does it produce?
3. Why does friction reduce efficiency?

Answers:

1. $$\frac{600}{800} \times 100 = 75\%$$
2. $$1000 - 250 = 750\text{ J}$$
3. Because friction changes useful mechanical energy into heat, which is less useful for the intended task.

17. Summary

Machine efficiency tells us how much of the input energy becomes useful output energy. It can be calculated using:

Any energy that does not become useful output is dissipated, usually as heat or sound. Friction and air resistance are common causes of this dissipation.

Even though useful energy decreases, total energy is still conserved. The input energy is always equal to the useful output energy plus the dissipated energy.