Newton's laws. Attnom Academy

S4.8.1 Chapter 1

Scalars and vectors

Mei walked 3 km east and then 3 km west. Did she go 6 km, or did she go nowhere?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A She went 6 km, and nothing else can be said.
You picked A. Read on. At the end, the chapter comes back to your guess.
B She went nowhere, so her journey was 0 km.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Both are right. They are two different quantities.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Sort quantities into scalars and vectors
Find the resultant of two forces along a line
Add two forces at right angles with a scale drawing

Some quantities need a number and a unit. Others need a direction too. This difference matters for every law in this book.

Mei walks 3 km east to the market. Then she walks 3 km west and stands at her own door. She walked a long way. Yet she is not away from home at all.

The two arrows are the two parts of the walk. Each has a size and a direction.

The distance she walked is 3 km + 3 km = 6 km. Distance counts every step, in any direction. It has a size and no direction.

The displacement is how far she is from where she began, in a straight line, and in which direction. Here it is 0 km. Her two walks cancel.

A quantity with a size and no direction is a scalar. A quantity with a size and a direction is a vector. Distance is a scalar. Displacement is a vector.

The first three rows pair a scalar with the vector that goes with it. The last two rows are one more scalar and one more vector. Weight is a force, and a force has a direction.

To give a vector, say its size, its unit and its direction: "20 N to the left". We draw it as an arrow. The length of the arrow shows the size. The head shows the direction.

We often pick one direction as positive, say right. A vector to the left then has a minus sign. The minus sign shows only the direction. A force of −15 N has the same size as +15 N and acts the other way.

Forces are vectors, so we add them with their directions. Start with forces on one line. Ravi pulls a box to the right with 300 N. Amina pulls it to the left with 200 N.

Draw it first

Two forces on one line

1 Ravi pulls to the right with 300 N. The arrow is 300 N long on the scale bar.
2 Amina pulls to the left with 200 N. Her arrow points the other way and is shorter.
3 Take 200 N from 300 N. One force of 100 N to the right does the work of both. It is the resultant.

The resultant is the single force that has the same effect as all the forces together. On one line, forces in the same direction add. Forces in opposite directions subtract. The resultant points the way of the bigger force.

If the two forces are equal and opposite, the resultant is 0 N. The forces are balanced.

See it three ways

As a picture

In words

On one line, forces in the same direction add. Forces in opposite directions subtract. The resultant points the way of the bigger force.

In symbols

400 N − 120 N = 280 N, to the right.

Worked example

Kofi pushes a cart to the right with 250 N. Lucía pushes it to the right with 140 N. Friction pushes it to the left with 120 N. Find the resultant force.

Choose right as the positive direction.
The forces to the right add: 250 N + 140 N = 390 N.
The force to the left is 120 N.
Resultant = 390 N − 120 N = 270 N, to the right.

Answer: 270 N to the right

Now put the forces at right angles. Hana pulls a sledge east with 30 N. Diego pulls it north with 40 N. East and north are at 90° to each other. The sledge goes neither east nor north.

We cannot add 30 N and 40 N as plain numbers. We draw them. Put the second arrow at the head of the first arrow, as two steps in a walk. The resultant runs from the start to the end.

The thick arrow is the resultant. A ruler on the grid shows it is 5 squares long.

Worked example

Hana pulls a sledge east with 30 N. Diego pulls it north with 40 N. Find the size of the resultant with a scale drawing.

Let 1 square stand for 10 N. The 30 N force is 3 squares. The 40 N force is 4 squares.
Draw the first arrow 3 squares east. From its head draw the second arrow 4 squares north.
Draw the resultant from the start to the end. Measure it: 5 squares.
Resultant = 5 squares × 10 N = 50 N.

Answer: 50 N

Pythagoras gives the same size without a drawing. The resultant is the longest side of a right angled triangle. So 30² + 40² = 900 + 1 600 = 2 500, and the square root of 2 500 is 50. A scale drawing is a check, and it also shows the direction. Here the resultant points about 53° north of east. A protractor on the drawing gives this angle.

At right angles the resultant is always less than the two sizes added. 50 N is less than 30 N + 40 N. Only forces on one line, in the same direction, add straight up.

Try it

Walk a right angle

You need: two small stones, a piece of string, about 8 steps long, flat open ground.

Be safe: Choose flat, open ground away from traffic. Walk slowly and watch where you step.

Put one stone down. This is your start.
Walk 4 steps straight ahead. Try to keep every step the same length.
Turn to your right, a quarter turn. Walk 3 steps.
Put the second stone down. Lay the string straight from the first stone to the second.
Count how many steps long the string is.

What you will see: You walked 4 + 3 = 7 steps. The string is only about 5 steps long. The string shows your displacement. It is the resultant of the two walks, and it is shorter than the sum.

If you cannot try it, this is what happens.

A wrong idea many people have

Forces of 30 N and 40 N always add up to 70 N.

Why it feels right: We add numbers all the time, and 30 + 40 is 70.

The test that breaks it: Draw the two forces at right angles on a grid, head to tail. The resultant measures 5 squares, which is 50 N. The walk in the Try it gives 7 steps walked and 5 steps from the start.

What is true: The sizes add to 70 N only when both forces act on one line in the same direction. At right angles, the resultant is the third side of a triangle, and it is shorter.

A wrong idea many people have

A displacement of −3 km means a shorter trip than +3 km.

Why it feels right: In everyday sums, −3 is less than +3.

The test that breaks it: Start at the door and walk 3 km east. Another day, start at the same door and walk 3 km west. Each walk ends 3 km from the door, on opposite sides.

What is true: The minus sign tells only the direction. Both walks have a size of 3 km. A displacement of −3 km is 3 km the other way.

Key words

distance: how far something travels in all, along its path. It is a scalar.
displacement: the straight line from the start to the end, with its direction. It is a vector.
scalar: a quantity with a size and no direction.
vector: a quantity with a size and a direction.
resultant: the one force that has the same effect as all the forces on a thing.

Back to your guess

The answer is C: Both are right. They are two different quantities.

Her distance walked is 6 km. Her displacement from the start is 0 km. Distance has a size only. Displacement has a size and a direction. Both are true at once.

Keep it

Copy this into your exercise book.

A scalar has a size only. A vector has a size and a direction. On one line, forces add or take away. At right angles they make a triangle: 30 N and 40 N give 50 N.

Draw: A right angled triangle of arrows, 3 squares and 4 squares, with the resultant of 5 squares.

S4.8.2 Chapter 2

Velocity and acceleration

A bus leaves the stop and picks up speed. How do we say how quickly it speeds up?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A By how much its velocity changes each second.
You picked A. Read on. At the end, the chapter comes back to your guess.
B By the distance it covers each second.
You picked B. Read on. At the end, the chapter comes back to your guess.
C By the top speed it reaches.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate acceleration from change in velocity and time
Find acceleration and distance from a velocity and time graph

In the last chapter you met vectors. Velocity is speed in a stated direction, so it is a vector. Its unit is the metre per second, m/s.

A car at 20 m/s east and a car at 20 m/s west have the same speed. They have different velocities. If east is positive, the second car has a velocity of −20 m/s.

Something accelerates when its velocity changes. It may speed up or slow down. It also accelerates when it turns, because a new direction is a new velocity. In this chapter we stay on a straight line.

The acceleration is the change in velocity divided by the time taken. We write u for the first velocity and v for the last one. Then a = (v − u) ÷ t, where t is the time in seconds.

A bus that gains 2 m/s every second has an acceleration of 2 m/s per second. We write this as 2 m/s², and say "two metres per second squared".

Draw it first

A bus gains 2 m/s every second

1 At 0 s the bus is at rest. After 1 s its velocity is 2 m/s. Mark both dots.
2 After 2 s it is 4 m/s, after 3 s it is 6 m/s and after 4 s it is 8 m/s. Mark each dot.
3 The dots lie on a straight line. Equal gains in velocity each second always make a straight line.

The steepness of the line is its gradient: the rise divided by the run. On this graph the rise is a change in velocity and the run is a time. So the gradient is (change in velocity) ÷ (time), and that is the acceleration.

Here the gradient is 8 m/s ÷ 4 s = 2 m/s². The unit follows from the axes: metres per second, divided by seconds, gives metres per second squared.

See it three ways

As a picture

In words

The velocity rises by the same amount each second, so the line is straight. The gradient of the line is the acceleration.

In symbols

a = (v − u) ÷ t = (8 − 0) ÷ 4 = 2 m/s²

Worked example

A car starts from rest and reaches 20 m/s in 8 s. Find its acceleration.

Write the rule: a = (v − u) ÷ t. Here u = 0 m/s, v = 20 m/s and t = 8 s.
The change in velocity is 20 m/s − 0 m/s = 20 m/s.
a = 20 m/s ÷ 8 s = 2.5 m/s².

Answer: 2.5 m/s²

The line rises 20 m/s in 8 s. Its gradient is the acceleration, 2.5 m/s².

When something slows down in the positive direction, its velocity falls, so the change in velocity is negative. The acceleration is then negative. We often call this deceleration. It means the acceleration points the opposite way to the positive direction.

Worked example

A cyclist slows from 12 m/s to 4 m/s in 4 s. Take her direction of travel as positive. Find her acceleration.

Here u = 12 m/s, v = 4 m/s and t = 4 s.
The change in velocity is 4 m/s − 12 m/s = −8 m/s.
a = −8 m/s ÷ 4 s = −2 m/s².

Answer: Acceleration −2 m/s²

A line that falls has a negative gradient. The cyclist is slowing down.

Now put the idea the other way round. If we know the acceleration, we can find the final velocity: v = u + at. A car at 5 m/s that accelerates at 2 m/s² for 4 s ends at 5 + 2 × 4 = 13 m/s.

A velocity time graph also gives distance. Take a steady velocity of 6 m/s for 10 s. The distance is 6 × 10 = 60 m. That is the area of the rectangle under the line: height 6 m/s times width 10 s.

The area between the line and the time axis is the displacement. While the velocity stays positive, the object never turns back, so the area is also the distance travelled. A triangle has area ½ × base × height.

A car speeds up, travels steadily, then brakes to rest. Each part has its own gradient and its own area.

Worked example

Find the distance the car travels in the 20 s of the graph above.

The distance is the area under the line. Split it into a triangle, a rectangle and a triangle.
From 0 s to 6 s the area is the triangle ½ × 6 × 12 = 36 m.
From 6 s to 16 s it is the rectangle 10 × 12 = 120 m.
From 16 s to 20 s it is the triangle ½ × 4 × 12 = 24 m.
Total distance = 36 m + 120 m + 24 m = 180 m.

Answer: 180 m

The same graph gives the accelerations. From 0 s to 6 s the gradient is 12 ÷ 6 = 2 m/s². From 6 s to 16 s the line is flat, so the acceleration is 0 m/s². From 16 s to 20 s the gradient is −12 ÷ 4 = −3 m/s².

When the acceleration is steady, the mean velocity is halfway between u and v. So the distance is ½ × (u + v) × t. This is the area of a trapezium. It works only when the acceleration is steady.

What if the velocity goes below zero? A trolley is pushed up a slope at 4 m/s. It slows, stops, then rolls back down. After 8 s it moves down the slope at 4 m/s. Take up the slope as positive.

The line crosses the time axis at 4 s. There the trolley stops for an instant, and then it moves back. The 5 s mark is hidden where the line crosses it.

The triangle above the axis is the trolley going up the slope: ½ × 4 × 4 = 8 m. The triangle below is it coming back: 8 m the other way. The displacement is 8 m − 8 m = 0 m. The trolley is back at its start.

The distance travelled is 8 m + 8 m = 16 m. So the area under a velocity time graph is the distance only while the velocity stays positive. Once it goes negative, the area below the axis is motion the other way.

The acceleration is steady all the time: (−4 − 4) ÷ 8 = −1 m/s². The trolley slows to a stop and speeds up the other way without any change in the acceleration.

Try it

Roll a ball down a slope

You need: a smooth board or flat plank, about 1.5 m long, a book to raise one end, a ball that rolls straight, small stones or chalk to mark places, a friend who can clap or tap a steady beat, about two each second.

Be safe: Do this on the floor, away from stairs and from people walking by. Keep the ball and stones away from small children.

Lay the board on the floor. Raise one end about 10 cm on the book. This is a gentle slope.
Hold the ball at the top. Your friend starts the beat. Let go on the first beat.
Roll it four times. On the first roll, mark where the ball is one beat after you let go. On the second roll, mark it two beats after, and so on.
Measure the gaps between the marks, one after another.
Do the whole set twice more and compare.

What you will see: Each gap is bigger than the one before. The gaps grow by about the same step each time. The ball covers more distance in each equal time, so its velocity is growing steadily. That is a steady acceleration.

If you cannot try it, this is what happens.

A wrong idea many people have

Acceleration is how fast something is going.

Why it feels right: In everyday speech, "accelerating" and "going fast" go together, and a fast car feels as if it has a lot of acceleration.

The test that breaks it: A car drives at a steady 30 m/s on a straight road. Its velocity does not change, so (v − u) ÷ t = 0 ÷ t = 0.

What is true: Acceleration is how quickly the velocity changes. A car at a very high, steady velocity has no acceleration. A car that starts from rest can have a large one.

A wrong idea many people have

A negative acceleration always means slowing down.

Why it feels right: "Deceleration" and a minus sign both sound like slowing, and in many first examples the sign is negative when something slows.

The test that breaks it: Take right as positive. A trolley moving left goes from −2 m/s to −8 m/s in 3 s. Its acceleration is (−8 − (−2)) ÷ 3 = −2 m/s². It is moving faster than before.

What is true: A negative acceleration points the negative way. If the object is moving the negative way too, it speeds up. It slows down only when the acceleration points against the velocity.

A wrong idea many people have

On a velocity time graph the height is the distance, and a flat line means the object is stopped.

Why it feels right: A distance time graph works that way, and the two graphs look alike.

The test that breaks it: Draw a flat line at 6 m/s for 10 s. The object moves 6 × 10 = 60 m, not 6 m. A stopped object is a line along the time axis, at 0 m/s.

What is true: On a velocity time graph the height is the velocity and the gradient is the acceleration. The distance is the area. A flat line above the axis is a steady velocity.

Key words

velocity: speed in a stated direction. It is a vector, in m/s.
acceleration: the change in velocity divided by the time taken, in m/s².
gradient: the steepness of a line: the rise divided by the run.
velocity time graph: a graph with time across and velocity up. The gradient is the acceleration and the area is the displacement.

Back to your guess

The answer is A: By how much its velocity changes each second.

How quickly the velocity changes is the acceleration. A bus that gains 2 m/s every second has an acceleration of 2 m/s². The distance and the top speed do not tell us that.

Keep it

Copy this into your exercise book.

Acceleration = (v − u) ÷ t, in m/s². On a velocity time graph the gradient is the acceleration and the area is the distance, while velocity stays positive.

Draw: A velocity time graph that rises, runs flat and falls, with the area split into a triangle, a rectangle and a triangle.

S4.8.3 Chapter 3

Newton's first law

When I stop pedalling, my bicycle slows down. Does a moving thing need a force to keep moving?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Yes. Without a force, a moving thing soon stops.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Yes, but only on a flat road.
You picked B. Read on. At the end, the chapter comes back to your guess.
C No. The bicycle slows because friction and the air push against it.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Explain why an object keeps moving unless a resultant force acts
Explain how seat belts protect passengers using inertia

For about two thousand years, many thinkers said that a moving thing needs a push to keep moving. It fits daily life. A cart stops when the ox stops pulling.

In the early 1600s, Galileo rolled balls down one slope and up another. The smoother the slopes, the nearer the ball rolled to its first height. He reasoned that on a perfectly smooth, flat track it would never stop.

Isaac Newton wrote this idea as the first of his three laws of motion. Here it is in modern words.

Newton’s first law: an object stays at rest, or keeps moving at a constant velocity, unless a resultant force acts on it. Constant velocity means the same speed and the same direction.

No force acts along the track. The trolley has moved on, and its velocity has not changed.

The two vertical forces are equal and opposite. Nothing pushes or pulls along the track.

Read the law with care. It does not say that no force acts. It says that no resultant force acts. Many forces can act, as long as they cancel. The weight and the support above cancel.

See it three ways

As a picture

In words

When the forces on an object are balanced, the resultant force is zero. The object then keeps its velocity.

In symbols

resultant = 30 N − 30 N = 0 N

A car cruises at a steady 20 m/s on a level road. The engine drives it forward. Air and the road drag it back. If the car holds a constant velocity, the first law says the resultant force is 0 N. So drive and drag are equal.

Only the forces along the road are drawn. The weight and the road’s upward push are equal and also cancel.

This is why a bicycle slows when you stop pedalling. The drive is gone, but drag and friction remain. The resultant now points backwards, so the velocity falls. Friction does the slowing. Nothing is needed to keep the bicycle moving.

In space a probe far from any planet keeps going with its engines off. Almost no force acts on it, so its velocity hardly changes. That is the first law at work.

The first law also says that an object at rest stays at rest. A stone on a table does not move by itself. It needs a resultant force to start moving.

Inertia is the tendency of an object to keep its velocity. The more mass an object has, the more inertia it has, and the harder it is to start, stop or turn. A loaded truck is harder to stop than an empty one.

Inertia explains seat belts. A car moves at 15 m/s with a passenger in it. The passenger has the same velocity as the car. Now the car hits a wall and stops in a very short time.

Draw it first

A car stops and the passenger does not

1 The car and the passenger both move at 15 m/s. No resultant force acts on the passenger, so the passenger keeps this velocity.
2 The car stops. The passenger has no force to slow them yet, so they keep moving forward at 15 m/s.
3 A seat belt pulls back on the passenger. This force slows the passenger to a stop along with the car.

Without a belt the passenger keeps moving until something stops them, such as the dashboard or the windscreen. That stop is sudden and the force is very large.

A seat belt acts on the passenger at once and slows them with the car. It is made to stretch a little, so it slows the passenger over a longer time. It also spreads the force over the strong bones of the chest and hips.

A headrest works the same way. When a car is hit from behind, the seat pushes the body forward. The head would stay behind because of its inertia. The headrest pushes it forward with the body.

Try it

Drop a stone into a cup

You need: a plastic or metal cup, a stiff piece of card, such as from a box, a small stone.

Be safe: Flick the card away from people and away from your face.

Stand the cup on a table. Lay the card on top of the cup.
Put the stone on the card, over the middle of the cup.
Flick the edge of the card sideways with one finger. Do it hard and fast.
Watch the stone.

What you will see: The card shoots out sideways. The stone drops straight down into the cup. The card was too quick to pull the stone along. Friction pulled on the stone for only a moment, so it hardly moved sideways. Then gravity pulled it down.

If you cannot try it, this is what happens.

A wrong idea many people have

A force is needed to keep an object moving.

Why it feels right: In daily life, things stop when we stop pushing. A cart, a ball and a bicycle all slow down.

The test that breaks it: Slide a stone on a rough floor, then on a smooth tiled floor. It stops sooner on the rough one. Friction is the force that stops it. The less friction there is, the further it goes.

What is true: No force is needed to keep a constant velocity. A resultant force is needed to change the velocity. Moving things slow down because friction and air resistance act on them.

A wrong idea many people have

If an object is at rest, no forces act on it.

Why it feels right: We notice forces when things move, and a still thing looks as if nothing is happening.

The test that breaks it: A book lies still on a table. Its weight pulls it down, and the table pushes it up. Take the table away and the book falls at once.

What is true: Forces can act on a thing at rest. They are balanced, so the resultant force is zero. The first law is about the resultant, not about each force.

A wrong idea many people have

When a car stops suddenly, a force throws the passengers forward.

Why it feels right: Passengers feel pushed into the belt, and they do move forward.

The test that breaks it: Put a stone on a smooth tray and carry the tray forward. Stop the tray suddenly. The stone slides on. No force pushed it forward. The tray stopped and the stone kept its velocity.

What is true: No forward force acts on the passengers. They keep the velocity they had, while the car slows under them. The belt gives the backward force that slows them.

Key words

first law: an object stays at rest, or keeps a constant velocity, unless a resultant force acts on it.
constant velocity: the same speed in the same direction.
balanced forces: forces whose resultant is zero.
inertia: the tendency of an object to keep its velocity. More mass means more inertia.

Back to your guess

The answer is C: No. The bicycle slows because friction and the air push against it.

A moving thing needs no force to keep its velocity. The bicycle slows because friction and air resistance act on it. A force is needed to change the velocity, not to keep it.

Keep it

Copy this into your exercise book.

First law: with no resultant force, an object at rest stays at rest and a moving one keeps a constant velocity. Inertia grows with mass. A seat belt gives the force that slows you with the car.

Draw: A car and passenger moving together, then the car stopped and the passenger still moving, with a belt arrow.

S4.8.4 Chapter 4

Newton's second law

An empty truck picks up speed quickly. A loaded truck with the same engine picks up speed slowly. Why?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A The same force gives a bigger mass a smaller acceleration.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The engine pushes harder when the truck is empty.
You picked B. Read on. At the end, the chapter comes back to your guess.
C On a flat road its weight pulls it backwards.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Use force equals mass times acceleration in calculations
Explain why a loaded truck speeds up more slowly than an empty one

The first law says that with no resultant force, the velocity stays the same. What happens when there is a resultant force? The velocity changes: the object accelerates. The second law says by how much.

Two things decide the acceleration. One is the resultant force. A bigger force on the same object gives a bigger acceleration. The other is the mass. The same force on a bigger mass gives a smaller acceleration.

Newton’s second law: the resultant force on an object equals its mass times its acceleration. In symbols, F = m × a. F is in newtons, N. m is in kilograms, kg. a is in metres per second squared, m/s².

The acceleration points in the same direction as the resultant force. A force to the right makes an object speed up if it moves right, and slow down if it moves left.

The newton is defined by this law. One newton is the resultant force that gives a mass of 1 kg an acceleration of 1 m/s². So 1 N = 1 kg m/s².

Take a trolley of mass 2 kg. We pull it with different resultant forces and measure its acceleration. Plot the results.

Draw it first

Acceleration against force for a 2 kg trolley

1 A force of 2 N gives an acceleration of 1 m/s². A force of 4 N gives 2 m/s². Mark both dots.
2 Forces of 6 N, 8 N and 10 N give accelerations of 3, 4 and 5 m/s². Mark each dot.
3 The dots lie on a straight line through the origin. Double the force and the acceleration doubles.

Now keep the force fixed at 12 N and change the mass. The more mass, the smaller the acceleration. Double the mass and the acceleration halves.

The same force of 12 N on a bigger mass gives a smaller acceleration.

We can rearrange F = m × a. To find the acceleration, divide the force by the mass: a = F ÷ m. To find the mass, divide the force by the acceleration: m = F ÷ a.

See it three ways

As a picture

In words

The resultant force on an object is its mass times its acceleration. The same force gives a bigger mass a smaller acceleration.

In symbols

a = F ÷ m = 20 N ÷ 4 kg = 5 m/s²

Always use the resultant force. If the engine pulls forward and drag pulls back, take one from the other first. Always use the mass in kilograms. A mass in grams must be divided by 1 000 first.

Worked example

A small car has a mass of 800 kg. Its engine gives a forward force of 2 400 N. Resistance pulls back with 800 N. Find its acceleration.

The resultant force is 2 400 N − 800 N = 1 600 N, forward.
Use a = F ÷ m, with F = 1 600 N and m = 800 kg.
a = 1 600 N ÷ 800 kg = 2 m/s², forward.

Answer: 2 m/s²

The arrows are to scale. The resultant is the long arrow minus the short one.

Worked example

A sprinter has a mass of 60 kg. She accelerates at 5 m/s² at the start of a race. What resultant force acts on her?

Use F = m × a, with m = 60 kg and a = 5 m/s².
F = 60 kg × 5 m/s² = 300 kg m/s².
One kg m/s² is one newton, so F = 300 N.

Answer: 300 N

The resultant force on the sprinter points the way she accelerates.

Now the trucks. An empty truck has a mass of 4 000 kg. Loaded, it has a mass of 12 000 kg. In both cases the resultant force is 6 000 N.

Worked example

A truck has a mass of 4 000 kg empty and 12 000 kg loaded. The resultant force is 6 000 N in both cases. Find each acceleration, and say how the two compare.

The resultant force is the same for both: 6 000 N.
Empty truck: a = 6 000 N ÷ 4 000 kg = 1.5 m/s².
Loaded truck: a = 6 000 N ÷ 12 000 kg = 0.5 m/s².
The loaded mass is 12 000 ÷ 4 000 = 3 times bigger, so the acceleration is 3 times smaller.

Answer: Empty 1.5 m/s², loaded 0.5 m/s²

The same resultant force of 6 000 N. The loaded truck has three times the mass and a third of the acceleration.

That answers the opening question. The same engine can give the same force whatever the load. A loaded truck has more mass, so by a = F ÷ m it speeds up more slowly. It also needs a bigger force than an empty truck to slow down at the same rate.

Try it

The same pull on a light box and a heavy box

You need: a small box with a string loop, a rubber band, stones to load the box, a smooth, level floor, a friend to count slow beats.

Be safe: Hold the band low and away from your face. A band that snaps or slips off can sting. Keep the floor clear so that nobody trips. Keep the stones away from small children.

Tie the rubber band to the string loop. Mark two dots on the band, 3 cm apart. When you pull, keep the dots 6 cm apart. This is your stretch.
Lay the empty box on the floor. Pull the band so that the two dots stay 6 cm apart. A friend counts two slow beats while you walk.
Mark where the box is after two beats. Measure how far it went from the start.
Now put the stones in the box. Pull with the same stretch for two beats and measure again.

What you will see: The same stretch gives about the same pull. The loaded box goes much less far in the same time. It has more mass, so the same force gives it a smaller acceleration. The stones also add friction, which makes the difference bigger.

If you cannot try it, this is what happens.

A wrong idea many people have

A bigger force always gives a bigger velocity.

Why it feels right: A harder push on a swing or a cart seems to make it go faster at once.

The test that breaks it: A car cruises at 20 m/s. The resultant force on it is 0 N, because the drive equals the drag. By F = m × a the acceleration is 0, yet the velocity is large.

What is true: The resultant force sets the acceleration, not the velocity. A steady resultant force makes the velocity grow steadily, for as long as it acts.

A wrong idea many people have

In F = m × a, the force is the push of the engine.

Why it feels right: The engine is the force we think about, and it is the biggest force in the picture.

The test that breaks it: The small car has an engine force of 2 400 N and resistance of 800 N. Using 2 400 N gives a = 3 m/s². The real acceleration is 2 m/s². Only the resultant, 1 600 N, gives that.

What is true: F is always the resultant force on the object. Add the forces first, with their directions. Then put the resultant into F = m × a.

A wrong idea many people have

You can put a mass in grams straight into F = m × a.

Why it feels right: Grams and kilograms are both units of mass, and the sum still looks correct.

The test that breaks it: A ball of 500 g has a resultant force of 2 N. Using 500 gives a = 2 ÷ 500 = 0.004. In kilograms, 500 g is 0.5 kg, and a = 2 ÷ 0.5 = 4 m/s².

What is true: The newton is defined with the kilogram. Change grams to kilograms first, by dividing by 1 000.

Key words

second law: the resultant force on an object equals its mass times its acceleration: F = m × a.
newton: the unit of force. One newton gives a mass of 1 kg an acceleration of 1 m/s².
resultant force: the one force that has the same effect as all the forces on an object together.

Back to your guess

The answer is A: The same force gives a bigger mass a smaller acceleration.

The engine gives the same force in both cases. The loaded truck has more mass, so the same resultant force gives it a smaller acceleration. This is Newton’s second law. A loaded truck does have a little more rolling friction, but the main reason is its mass.

Keep it

Copy this into your exercise book.

Second law: resultant force = mass × acceleration, F = m × a. Use the resultant, and the mass in kg. Double the mass and the acceleration halves.

Draw: A box with a big arrow and a small arrow, and the sum a = F ÷ m beside it.

S4.8.5 Chapter 5

Weight and falling

A skydiver jumps from a plane. Why does she stop speeding up long before she lands?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Gravity switches off when she is falling fast.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The air pushes up on her harder and harder, until it equals her weight.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Her mass gets smaller as she falls.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate weight from mass and gravitational field strength
Explain terminal velocity using the forces on a falling object
Time paper cones falling to show terminal velocity

Weight is a force. It is the pull of gravity on an object, so it is a vector that points down. We measure it in newtons. Mass is the amount of matter in an object. We measure it in kilograms.

Each kilogram feels a pull from gravity. The pull on each kilogram is the gravitational field strength, g. Its unit is the newton per kilogram, N/kg.

At the surface of the Earth, g is about 9.8 N/kg. It is a little more near the poles and a little less near the equator, from about 9.78 to 9.83 N/kg. Some books and quick sums use 10 N/kg. This book uses 9.8 N/kg, so answers with 10 will differ a little.

The weight of an object is its mass times g: W = m × g. W is in newtons, m in kilograms and g in N/kg.

See it three ways

As a picture

In words

Weight is the pull of gravity on an object. It is a force, so it is measured in newtons. Mass is not a force.

In symbols

W = m × g = 2 kg × 9.8 N/kg = 19.6 N

Gravitational field strength at the surface. For Jupiter the value is at its cloud tops, about 25 N/kg.

Worked example

A bag has a mass of 5 kg. Find its weight on Earth, using g = 9.8 N/kg.

Use W = m × g, with m = 5 kg and g = 9.8 N/kg.
W = 5 × 9.8 = 49 N.
The weight is 49 N, acting downwards.

Answer: 49 N

The bag hangs still, so the rope pulls up with a force equal to the weight.

Mass stays the same wherever you go. Weight changes with g. An astronaut who goes to the Moon has the same mass and a smaller weight, because g on the Moon is smaller.

Worked example

An astronaut has a mass of 80 kg. Find her weight on Earth and on the Moon, where g is 1.6 N/kg.

Her mass is 80 kg in both places. Only g changes.
On Earth: W = 80 × 9.8 = 784 N.
On the Moon: W = 80 × 1.6 = 128 N.

Answer: Earth 784 N, Moon 128 N

The same 80 kg of astronaut. The weight on the Moon is much smaller.

Now let an object fall with nothing else acting but its weight. By the second law, a = W ÷ m = (m × g) ÷ m = g. So every object falls with the same acceleration, g = 9.8 m/s². The unit N/kg is the same as m/s².

In 1971 an astronaut on the Moon, where there is no air, dropped a hammer and a feather together. They reached the ground at the same time. On Earth a feather falls slowly because of the air.

Air pushes up on a falling object. We call this force drag, or air resistance. Drag grows as the speed grows. It also depends on the shape and the area of the object. Now follow a skydiver of mass 80 kg, whose weight is 784 N.

The weight never changes. The drag grows with speed until it equals the weight.

At the first moment she is hardly falling yet, so there is no upward drag. The resultant is her whole weight, and her acceleration is g, 9.8 m/s².

As she speeds up, the drag grows. The resultant force down gets smaller, so her acceleration gets smaller. She still speeds up, but more slowly.

At last the drag equals the weight. The resultant is 0 N, and her acceleration is 0. She falls at a constant velocity. This is the terminal velocity. A skydiver falling flat and spread out has a terminal velocity of about 55 m/s. It differs from person to person.

Her weight has not changed, and she has not stopped. Gravity still pulls her down. She keeps falling at a constant velocity, because the forces are balanced.

A model of a jump, not measured data. The gradient shrinks to zero at the first terminal velocity. The parachute opens at 20 s.

When the parachute opens, the area facing the air is much bigger. The drag jumps above the weight. The resultant force now points up, so she slows down.

As she slows, the drag falls, until it equals the weight again. This is a new, smaller terminal velocity.

As she slows, the drag falls. She stops slowing when the drag again equals the weight. Then she falls at a new, smaller terminal velocity, about 5 m/s with an open parachute. A smooth landing is possible at that speed.

Worked example

A skydiver of mass 80 kg falls at her terminal velocity. Her parachute then opens and the drag jumps to 1 568 N. Find the drag before it opens, and her acceleration just after.

At terminal velocity the resultant force is zero, so the drag equals the weight.
The weight is 80 kg × 9.8 N/kg = 784 N, so the drag was 784 N.
Just after the parachute opens, the resultant is 1 568 N − 784 N = 784 N, upwards.
Her acceleration is 784 N ÷ 80 kg = 9.8 m/s², upwards.

Answer: Before: 784 N. After: 9.8 m/s² upwards

Try it

Time a paper cone as it falls

You need: a sheet of paper, torn into a square, tape, or fold the ends in, a string stuck up a wall, marked at 1 m, 1.5 m and 2 m above the floor, a clock or phone that shows tenths of a second, a friend to time.

Be safe: Stand on the floor and hold the cone up high. Never drop things from a window, a roof or a balcony, and do not stand on a chair.

Roll the paper square into a cone with a point. Tape the edge or tuck it in.
Hold the cone point down, level with the 1 m mark on the string. Let go. Your friend times it from the moment you let go until it lands.
Drop it five times and take the middle time. Then do the same from 1.5 m and from 2 m.
Write down height and time for each. Plot time across and height up.

What you will see: Each extra half metre takes about the same extra time. The cone fell its first stretch while speeding up. Then the drag grew to equal its weight, and it fell at a constant velocity. The gradient of the line is the terminal velocity.

If you cannot try it, this is what happens.

In the example, 0.5 m more takes 0.3 s more. The terminal velocity is about 0.5 ÷ 0.3, which is 1.7 m/s. Your results will differ, but the later part of the line should be straight.

A wrong idea many people have

Mass and weight are the same thing.

Why it feels right: People say "my weight is 60 kilograms", and a kitchen scale marks its dial in kilograms.

The test that breaks it: Take a 60 kg person to the Moon. Her mass is still 60 kg, since she is made of the same matter. Her weight falls from about 588 N to 96 N, because g is smaller there.

What is true: Mass, in kilograms, is how much matter there is, and it does not change with place. Weight, in newtons, is the pull of gravity and it does. Weight = mass × g.

A wrong idea many people have

Heavier objects always fall faster.

Why it feels right: A stone falls faster than a leaf, and a leaf is lighter.

The test that breaks it: Where there is no air, a hammer and a feather fall together. Screw a sheet of paper into a tight ball and drop it with a flat sheet. The ball falls faster, with the same weight.

What is true: With no drag every object falls with the same acceleration, g. With drag, the shape and area matter. A flat sheet has more drag than a ball of the same weight.

A wrong idea many people have

At terminal velocity the skydiver has stopped, or her weight has gone.

Why it feels right: She stops speeding up, and "terminal" sounds like the end.

The test that breaks it: Look at the forces. The weight is still 784 N down. The drag is 784 N up. The resultant is zero, so her velocity is constant, and it is still large.

What is true: At terminal velocity she keeps falling at a constant velocity. The weight is the same, and the drag has grown to equal it.

Key words

weight: the pull of gravity on an object, in newtons. W = m × g.
gravitational field strength: the force of gravity on each kilogram, g, in N/kg. It is 9.8 N/kg on Earth.
drag: the force of air or water pushing against a moving object.
terminal velocity: the constant velocity of a falling object when the drag equals its weight.

Back to your guess

The answer is B: The air pushes up on her harder and harder, until it equals her weight.

The air pushes up harder as she speeds up. When that push equals her weight, the resultant force is zero. She then falls at a constant velocity, called the terminal velocity.

Keep it

Copy this into your exercise book.

Weight W = m × g, in newtons, with g = 9.8 N/kg on Earth. Mass does not change. A falling object has terminal velocity when drag = weight and the resultant is zero.

Draw: A falling ball with a weight arrow down and a drag arrow up growing until they are equal.

S4.8.6 Chapter 6

Newton's third law

If every push has an equal push back, how can anything ever start to move?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A It cannot. Equal and opposite forces always cancel.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The bigger object always wins.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The two forces act on different objects, so they do not cancel.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Identify force pairs in walking, swimming and rockets
Explain why the two forces in a pair act on different objects

When you push a wall, the wall seems to do nothing. In fact it pushes back on you. Forces always come in pairs.

Newton’s third law: if object A exerts a force on object B, then B exerts a force on A. The two forces are equal in size, opposite in direction, and of the same type. They act at the same time.

Take the four parts one at a time. Size: the two forces have the same number of newtons. Direction: they point opposite ways. Type: both are contact forces, or both are gravity, or both are electric, or both are magnetic. Place: each acts on a different object.

Name each force by what pushes and what is pushed. One is the "force of the hand on the wall". The other is the "force of the wall on the hand". The two names swap the objects. We call the two forces a force pair.

Each arrow is named for what pushes. The two arrows are the same length and point opposite ways. They are on different blocks.

The forces of a pair do not cancel. To cancel, two forces must act on the same object. The force of the hand acts on the wall. The force of the wall acts on the hand. One object has only one of the two.

See it three ways

As a picture

In words

When a bat hits a ball, each pushes the other. The two forces are equal in size and opposite in direction, and each acts on a different object.

In symbols

force of bat on ball = force of ball on bat = 200 N

Now walking. You walk by pushing the ground backward with your foot. The ground pushes your foot forward. That forward push is a friction force, and it is what moves you.

The foot pushes the ground back. The ground pushes the foot forward with a force of the same size.

On smooth ice there is almost no friction. Your foot cannot push the ground back with much force, so the ground cannot push you forward. You slip.

A swimmer pushes the water backward with her hands and feet. The water pushes her forward. A boat does the same with its oars or propeller.

The hand pushes the water back. The water pushes the hand, and the swimmer, forward.

A rocket pushes hot gas down and out of its engine. The gas pushes the rocket up. The rocket does not push on the air or the ground. So it works in empty space, where there is nothing to push against.

The rocket pushes the gas down. The gas pushes the rocket up with a force of the same size.

Worked example

A small rocket has a mass of 20 kg. Its engine pushes gas down with 300 N. Find the acceleration of the rocket at lift-off. Use g = 9.8 N/kg.

By the third law, the gas pushes the rocket up with 300 N, the same as the rocket pushes the gas down.
The weight is 20 kg × 9.8 N/kg = 196 N, down.
The resultant is 300 N − 196 N = 104 N, up.
a = 104 N ÷ 20 kg = 5.2 m/s², up.

Answer: 5.2 m/s²

The two forces of a pair are always equal. The accelerations they cause are not equal if the masses differ. Each acceleration is a = F ÷ m for its own object.

Worked example

Skater A has a mass of 40 kg. Skater B has a mass of 60 kg. They stand on smooth ice and push each other apart with a force of 120 N. Find each acceleration.

The forces are a pair, so each skater feels 120 N, in opposite directions.
Skater A: a = 120 N ÷ 40 kg = 3 m/s².
Skater B: a = 120 N ÷ 60 kg = 2 m/s².
The lighter skater accelerates more. Both forces are the same.

Answer: A 3 m/s², B 2 m/s²

The arrows are the same. The skaters have different masses, so their accelerations differ.

The same idea explains why the Earth does not seem to move when you jump. The Earth pulls you down with your weight. You pull the Earth up with a force of the same size. The Earth has so much mass that its acceleration is far too small to notice.

Now a trap. A book lies on a table. The weight of the book pulls it down. The table pushes it up. These forces are equal and opposite. Are they a third law pair? They are not.

Both forces act on the book. They are balanced forces, so they are not a pair.

They act on the same object, the book. They are different types, because weight is gravity and the push of the table is a contact force. A pair has the forces on two objects, of one type.

The real pairs are these. The push of the table on the book pairs with the push of the book on the table, both contact forces. The pull of the Earth on the book pairs with the pull of the book on the Earth, both gravity.

The push of the table on the book, and the push of the book on the table, are a real pair. They act on two objects.

There is a simple test. Take the table away. The weight stays, but the push of the table is gone. The two forces of a real pair appear and vanish together.

Try it

Two rubber bands pulling

You need: two rubber bands of the same kind, a friend, a ruler.

Be safe: Stretch the bands gently, with both hands, and never let go of a stretched band. A band can snap back and sting. Keep it away from your face and from other people’s faces.

Join the two bands in a chain by looping one through the other.
You hold the free end of one band. A friend holds the free end of the other.
Pull gently apart until the bands are stretched. Measure the length of each band.
Now swap sides, and let the stronger of you pull harder. Measure again.

What you will see: The two bands stretch to the same length, however hard each of you tries. Each of you feels the band pull back on your hand. That pull equals the pull of your hand on the band. A band can pull on you only as hard as you pull on it.

If you cannot try it, this is what happens.

A wrong idea many people have

Equal and opposite forces always cancel, so nothing can ever start to move.

Why it feels right: Balanced forces do cancel, and the words "equal and opposite" sound the same.

The test that breaks it: Two skaters push each other apart. The force on skater A is 120 N one way. The force on skater B is 120 N the other way. Neither skater feels both forces, and both skaters move.

What is true: Two forces cancel only if they act on the same object. The two forces of a pair act on different objects, so each object has an unbalanced force on it and accelerates.

A wrong idea many people have

In a collision, the bigger object pushes harder on the smaller one.

Why it feels right: A truck does far more damage to a fly than the fly does to the truck.

The test that breaks it: Look at the pair of forces in a crash between a truck and a fly. The force of the truck on the fly and the force of the fly on the truck are equal. Their size is the same, and the directions are opposite.

What is true: A pair of forces is always equal. The fly is thrown aside because its mass is tiny, so a = F ÷ m is huge. The truck has a huge mass, so its acceleration is tiny.

A wrong idea many people have

The weight of a book and the push of the table on it are a third law pair.

Why it feels right: They are equal and opposite, which is what the third law says.

The test that breaks it: Both forces act on the book. One is gravity, and the other is a contact force. The weight stays if the table is taken away, and the push does not.

What is true: A pair is two forces of the same type, on two different objects. The push of the table on the book pairs with the push of the book on the table.

Key words

third law: if A exerts a force on B, then B exerts a force of equal size and opposite direction on A.
force pair: the two forces of the third law. They have the same size and type, opposite directions, and act on different objects.
contact force: a force between two things that touch, such as a push, friction or the push of a table.

Back to your guess

The answer is C: The two forces act on different objects, so they do not cancel.

The two forces of a pair act on different objects. Forces cancel only when they act on the same object, so a pair never cancels itself.

Keep it

Copy this into your exercise book.

Third law: the two forces of a pair are equal, opposite, of the same type, and on different objects. They never cancel. A weight and the push of a table on a book are not a pair.

Draw: Two blocks, each with one arrow of the same length, pointing opposite ways, labelled "A on B" and "B on A".

S4.8.7 Chapter 7

Momentum

Why do cars have parts built to crumple in a crash, when a stiff car would stay in one piece?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A A crumpling car takes longer to stop, so the force on the passengers is smaller.
You picked A. Read on. At the end, the chapter comes back to your guess.
B A crumpling car stops sooner, so the passengers feel less.
You picked B. Read on. At the end, the chapter comes back to your guess.
C A crumpling car has less momentum to start with.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Calculate momentum and use its conservation in collisions
Explain how crumple zones and helmets reduce force by increasing time

A truck moving slowly is hard to stop. A fast football is also hard to stop, in its own way. Both mass and velocity matter. Momentum puts the two together.

Momentum is mass times velocity: p = m × v. The mass is in kilograms and the velocity is in m/s. The unit of momentum is the kilogram metre per second, kg m/s.

Momentum is a vector. It points the way the velocity points. Take right as positive. A 2 kg ball moving left at 5 m/s has a momentum of 2 × (−5) = −10 kg m/s. The minus sign shows it is moving left.

A car of mass 1 000 kg moving at 20 m/s has a momentum of 1 000 × 20 = 20 000 kg m/s. A 0.16 kg cricket ball moving at 25 m/s has a momentum of 0.16 × 25 = 4 kg m/s.

Now two things collide. One trolley hits another on a smooth track. Each pushes the other with a force. By the third law the two forces are equal and opposite, and they act for the same time.

We will see that a force acting for a time changes momentum. The two trolleys get equal and opposite changes of momentum. One gains what the other loses. So the total does not change. This is the conservation of momentum.

Conservation of momentum: if no outside resultant force acts on a group of objects, their total momentum stays the same. It is equal before and after a collision. Add momenta with their signs.

See it three ways

As a picture

In words

The total momentum before a collision equals the total momentum after it. One object can gain what another loses.

In symbols

1 × 9 + 2 × 0 = (1 + 2) × 3 = 9 kg m/s

Friction and other outside forces always exist a little. In a quick collision, the forces between the objects are far bigger than they are, so the total momentum is nearly unchanged.

Worked example

Trolley A has a mass of 2 kg and moves right at 3 m/s. It hits trolley B, of mass 1 kg, which is at rest. They stick together. Find their velocity after.

Take right as positive. Before: A has 2 kg × 3 m/s = 6 kg m/s. B has 0 kg m/s.
The total momentum before is 6 + 0 = 6 kg m/s.
After, the trolleys move together. Their mass is 2 + 1 = 3 kg, at a velocity v.
Momentum after = 3 × v. It equals 6, so v = 6 ÷ 3 = 2 m/s, to the right.

Answer: 2 m/s to the right

The total momentum is 6 kg m/s before and 6 kg m/s after.

Take care with signs. When objects move towards each other, one has a negative momentum. The total may be smaller than either.

Worked example

Trolley A has a mass of 3 kg and moves right at 4 m/s. Trolley B has a mass of 1 kg and moves left at 2 m/s. They collide and stick together. Find their velocity after.

Take right as positive. B moves left, so its velocity is −2 m/s.
Before: 3 × 4 + 1 × (−2) = 12 − 2 = 10 kg m/s.
After, the mass is 3 + 1 = 4 kg, so 4 × v = 10.
v = 10 ÷ 4 = 2.5 m/s, to the right.

Answer: 2.5 m/s to the right

The total momentum is 10 kg m/s before and after. The trolley moving left took some away.

Conservation of momentum also works when things push apart. Two trolleys at rest have a total momentum of 0. A spring between them pushes them apart. After the push the total must still be 0.

Worked example

Trolley A has a mass of 2 kg and trolley B has a mass of 1 kg. They are at rest with a compressed spring between them. The spring pushes them apart. A leaves to the left at 3 m/s. Find the velocity of B.

Take right as positive. Before the push, both are at rest, so the total momentum is 0.
After: A has 2 kg × (−3 m/s) = −6 kg m/s.
The total must stay 0, so B has a momentum of +6 kg m/s.
B has a mass of 1 kg, so v = 6 ÷ 1 = 6 m/s, to the right.

Answer: 6 m/s to the right

The momenta are −6 kg m/s and +6 kg m/s. They add to 0, as before the push.

Note what is conserved. The total momentum is conserved. The momentum of each trolley is not. A went from 0 to −6 kg m/s. B went from 0 to +6 kg m/s. In a sticking collision some energy of movement turns into heat and sound. So the energy of movement is not conserved, but the momentum is.

Now the force. The second law says F = m × a. The acceleration is the change in velocity divided by the time, a = (v − u) ÷ t. So F = m × (v − u) ÷ t = (m × v − m × u) ÷ t.

The top line is the final momentum minus the first momentum, the change in momentum. So the resultant force is the change in momentum divided by the time taken: F = Δp ÷ t. This is the second law in the form Newton first wrote it.

The same change of momentum can happen quickly or slowly. If it takes a longer time, the force is smaller. This is the idea behind many safety devices.

A car of mass 1 000 kg moving at 20 m/s hits a wall and stops. Its momentum changes by 20 000 kg m/s. The numbers below are round ones to show the idea. Real crashes vary.

Draw it first

The same change of momentum, a different time

1 A stiff car stops in 0.025 s. The force is 20 000 ÷ 0.025 = 800 000 N, which is 800 kN.
2 A crumple zone crushes slowly and the stop lasts 0.1 s. The force is 20 000 ÷ 0.1 = 200 000 N, which is 200 kN.

Worked example

A car of mass 1 000 kg moving at 20 m/s hits a wall and stops. Find the average force if the stop takes 0.025 s, and if a crumple zone makes it take 0.1 s.

The car goes from 20 m/s to rest, so its change in momentum is the whole of its momentum.
The momentum is 1 000 × 20 = 20 000 kg m/s.
Stiff car: F = 20 000 ÷ 0.025 = 800 000 N.
Crumple zone: F = 20 000 ÷ 0.1 = 200 000 N.
The time is 0.1 ÷ 0.025 = 4 times longer, so the force is 4 times smaller.

Answer: Stiff 800 000 N, crumple zone 200 000 N

A bicycle helmet works the same way. Treat the head as a lump of 5 kg. It meets hard ground at 6 m/s. Its momentum is 5 × 6 = 30 kg m/s. On bare ground it stops in about 0.002 s, so the force is 30 ÷ 0.002 = 15 000 N.

A helmet has foam that crushes. The stop lasts about 0.008 s. The force is 30 ÷ 0.008 = 3 750 N. The head loses the same momentum, with a quarter of the force. A real head is more complex than this model, but the idea is right.

The same change of momentum, 30 kg m/s. The foam makes the stop four times longer, so the force is four times smaller.

Airbags, seat belts that stretch and padded playground floors do this. So does bending your knees when you land. Each spreads the same change of momentum over a longer time.

Try it

Catch a soft ball two ways

You need: a rolled up sock, a friend.

Be safe: Toss the sock gently, at the hands and never at a face. Use only something soft.

Stand a short distance from your friend. Ask your friend to toss the sock gently at your hands.
Catch it with stiff, still hands. Notice how it feels.
Catch it again, but let your hands move back with the sock as you catch it. Notice how it feels.
Do each way a few times.

What you will see: Catching with hands that move back feels gentler. The sock stops over a longer time, so the force on your hands is smaller. The momentum the sock loses is the same both ways.

If you cannot try it, this is what happens.

A wrong idea many people have

Momentum is always positive, so it only needs a size.

Why it feels right: Mass and speed are never negative, and a product of them looks as if it should be positive.

The test that breaks it: Two equal trolleys move towards each other at the same speed and stick together. They stop. The total momentum before was +p and −p, which is 0, and 0 is also the total after.

What is true: Momentum is a vector. Moving the other way gives a negative momentum. Always choose a positive direction and write the minus sign.

A wrong idea many people have

In a collision, the momentum of each object stays the same.

Why it feels right: The rule says momentum is conserved, and it is easy to read this as true for each object.

The test that breaks it: In the spring example, A goes from 0 to −6 kg m/s and B goes from 0 to +6 kg m/s. Each changed.

What is true: Only the total momentum of the whole group is conserved. One object can gain what another loses.

A wrong idea many people have

A crumple zone or a helmet reduces the momentum the person loses.

Why it feels right: Safety devices "take away" force, so it seems they also take away momentum.

The test that breaks it: In the crash, the car and the passengers go from 20 m/s to rest, with or without a crumple zone. The change of momentum is the same: 20 000 kg m/s for the car.

What is true: The change of momentum is the same. A crumple zone or a helmet makes the stop last longer, and F = Δp ÷ t then gives a smaller force.

Key words

momentum: mass times velocity, in kg m/s. It is a vector.
conservation of momentum: with no outside resultant force, the total momentum before a collision equals the total after.
crumple zone: a part of a car built to crush slowly in a crash, so the stop lasts longer.

Back to your guess

The answer is A: A crumpling car takes longer to stop, so the force on the passengers is smaller.

The passengers lose the same momentum either way. Crumpling makes the stop last longer. The same change of momentum over a longer time needs a smaller force.

Keep it

Copy this into your exercise book.

Momentum p = m × v, in kg m/s, with a sign for direction. Total momentum is conserved in a collision. Force = change of momentum ÷ time, so a longer stop gives a smaller force.

Draw: Two trolleys before and after a collision, with their momenta added. Beside them, bars for the force in a short stop and a long stop.

S4.8.8 Chapter 8

Stopping distances

A child steps into the road. Why can a car not stop the moment the driver sees her?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A The brakes take a long time before they start to work.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The car keeps moving while the driver reacts, and again while the brakes slow it.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Only the road decides how far a car goes before it stops.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Explain what affects thinking distance and braking distance
Use a measured reaction time to work out thinking distance

A driver sees danger, thinks, and presses the brake. The car then slows and stops. Both parts take time, and the car keeps moving for all of it.

The thinking distance is how far the car travels while the driver reacts, before the brakes start to work. The braking distance is how far it travels from the moment the brakes act until it stops.

The stopping distance is the two added together: stopping distance = thinking distance + braking distance.

See it three ways

As a picture

In words

The car travels on while the driver reacts. Then it travels on while the brakes work. The two distances add up to the stopping distance.

In symbols

10.5 m + 22.5 m = 33 m

The reaction time is the time from seeing the danger to pressing the brake. During it the car keeps its speed. So thinking distance = speed × reaction time.

Speeds on roads are often in km/h. For this sum, change them to m/s. One km/h is 1 000 m in 3 600 s, so divide by 3.6. A car at 72 km/h is at 72 ÷ 3.6 = 20 m/s.

How long is a reaction time? A simple test, such as catching a falling ruler, gives about 0.15 s to 0.25 s. A driver has to see, decide and act, and this takes longer, often about 0.7 s.

You can measure your own simple reaction time with a ruler. Your friend holds the ruler upright and drops it. You catch it. The ruler falls from rest with the acceleration g, 9.8 m/s². The distance it falls tells you the time.

On a velocity time graph the ruler’s line is v = g × t. The distance fallen is the area of the triangle: d = ½ × t × (g × t) = ½ × g × t². Turn this round to get t = √(2d ÷ g).

Worked example

A ruler falls 20 cm before Hana catches it. Find her reaction time. Use g = 9.8 m/s².

The ruler falls from rest, so d = ½ × g × t², and so t = √(2d ÷ g).
Change the distance to metres: 20 cm = 0.2 m.
t = √(2 × 0.2 ÷ 9.8) = √0.0408 ≈ 0.20 s.

Answer: 0.2 s

After 0.2 s the ruler moves at 1.96 m/s. The triangle has area ½ × 0.2 × 1.96 = 0.196 m, about 20 cm.

Try it

Catch a falling ruler

You need: a ruler, or a long strip of card marked in centimetres, a friend.

Be safe: Let the ruler drop. Do not throw it. Keep your face away from the falling end.

Your friend holds the ruler upright by its top end. Hold your thumb and finger open, either side of the bottom end, with the 0 mark level with them.
Your friend lets go without warning. Close your fingers as soon as you see it fall.
Read the mark at the top of your thumb. This is the distance the ruler fell, in centimetres.
Do it five times and take the usual value. Look up the time in the table.

What you will see: Most people catch the ruler after 10 cm to 30 cm. That is a time of about 0.14 s to 0.25 s. A driver reacts more slowly, because there is more to decide than a ruler falling.

If you cannot try it, this is what happens.

Now use a reaction time for a driver. Thinking distance = speed × reaction time. At twice the speed, the car goes twice as far in the same time. So thinking distance grows in proportion to speed.

Worked example

Lucía drives at 90 km/h. Her reaction time is measured as 0.8 s. Find her thinking distance.

Change the speed to m/s: 90 km/h ÷ 3.6 = 25 m/s.
The car keeps this speed while she reacts, so thinking distance = speed × time.
Thinking distance = 25 m/s × 0.8 s = 20 m.

Answer: 20 m

The car travels 20 m before the brakes start to act.

Several things make the reaction time longer. These are tiredness, alcohol, some medicines and drugs, and distractions such as a phone. A longer reaction time means a longer thinking distance, at any speed.

Now the braking distance. The brakes give a braking force, and by the second law a = F ÷ m. We call a steady slowing a deceleration. Take a car that slows by 5 m/s every second, a deceleration of 5 m/s².

From 20 m/s the car needs 20 ÷ 5 = 4 s to stop. The graph is a straight line down. So the braking distance is the area of a triangle: ½ × 4 × 20 = 40 m.

Now try 10 m/s, then 30 m/s, with the same deceleration. From 10 m/s the car stops in 2 s. From 30 m/s it takes 6 s. The graph shows the three triangles.

Same deceleration, so the same gradient. A higher starting speed makes a taller and wider triangle.

The areas are ½ × 2 × 10 = 10 m, ½ × 4 × 20 = 40 m and ½ × 6 × 30 = 90 m. Double the speed from 10 to 20 m/s, and the braking distance goes from 10 m to 40 m. It is four times as big, because both the base and the height of the triangle double.

So the braking distance grows with the square of the speed. This holds whenever the deceleration is the same at every starting speed.

Worked example

A car travels at 72 km/h. The driver’s reaction time is 0.7 s. The car brakes with a deceleration of 5 m/s². Find the stopping distance.

Change the speed to m/s: 72 km/h ÷ 3.6 = 20 m/s.
Thinking distance = 20 m/s × 0.7 s = 14 m.
The car needs 20 ÷ 5 = 4 s to stop, so the braking distance is ½ × 4 × 20 = 40 m.
Stopping distance = 14 m + 40 m = 54 m.

Answer: 54 m

The stopping distance is the whole bar, 54 m.

The bars below show the three speeds. The thinking part grows in step with the speed. The braking part grows much faster.

With a reaction time of 0.7 s and a deceleration of 5 m/s². Thinking distances are 7, 14 and 21 m. Braking distances are 10, 40 and 90 m.

What changes the braking distance? First, the speed, as shown. Second, the grip between tyres and road. Third, the condition of the brakes and tyres. Fourth, the mass of the vehicle. Fifth, the slope of the road.

On a wet or icy road the tyres grip less. The braking force is smaller, so the deceleration is smaller, and the car takes longer to stop. Worn tyres and worn brakes do the same.

With the same braking force, a heavier vehicle has a smaller deceleration, since a = F ÷ m. So it takes longer to stop. This is why trucks have much bigger brakes. Going downhill also makes the braking distance longer.

Take the 20 m/s car again. Suppose the road is wet and the deceleration is only 2.5 m/s², half as much. The car needs 8 s to stop, and the braking distance is ½ × 8 × 20 = 80 m. This is twice the dry distance. The values are round numbers for the example.

The thinking distance is the same. The wet road doubles the braking distance.

Note which part each cause changes. Tiredness, alcohol and a phone change the thinking distance. Wet roads, worn tyres, a heavy load and a slope change the braking distance. Speed changes both.

A wrong idea many people have

The stopping distance is how far the car goes after the brakes are pressed.

Why it feels right: We usually think of stopping as the part we feel, with the brakes on and the car slowing.

The test that breaks it: A car at 20 m/s and a reaction time of 0.7 s travels 14 m before the brakes do anything. A person 45 m away is beyond the braking distance of 40 m. Yet the car travels 14 m + 40 m = 54 m, so it reaches them.

What is true: The stopping distance is the thinking distance plus the braking distance. The car moves for the whole of the thinking time, with no braking.

A wrong idea many people have

If the speed doubles, the stopping distance doubles.

Why it feels right: Distance and speed go together, and thinking distance does double with speed.

The test that breaks it: From 10 m/s to 20 m/s the thinking distance goes from 7 m to 14 m, which is double. The braking distance goes from 10 m to 40 m, which is four times. The stopping distance goes from 17 m to 54 m, about three times.

What is true: Thinking distance is in proportion to speed. Braking distance grows with the square of the speed. So the stopping distance grows faster than the speed.

A wrong idea many people have

A tired driver needs a longer braking distance.

Why it feels right: A tired driver stops later, so it feels as if the braking has gone wrong.

The test that breaks it: Take a car at 20 m/s. With a reaction time of 0.7 s the thinking distance is 14 m. With 1.5 s it is 30 m. The braking distance is 40 m in both cases, since the brakes and the road are the same.

What is true: Reaction time changes only the thinking distance. The braking distance depends on the speed, the grip, the brakes and the load.

Key words

reaction time: the time from seeing danger to pressing the brake.
thinking distance: the distance the car travels during the reaction time. It is speed × reaction time.
braking distance: the distance the car travels while the brakes slow it to rest.
stopping distance: the thinking distance plus the braking distance.

Back to your guess

The answer is B: The car keeps moving while the driver reacts, and again while the brakes slow it.

There are two stretches. First the car travels on while the driver reacts. Then it travels on while the brakes slow it. The two distances add up to the stopping distance.

Keep it

Copy this into your exercise book.

Stopping distance = thinking distance + braking distance. Thinking distance = speed × reaction time. Braking distance grows with the square of the speed. Wet roads and heavy loads make it longer.

Draw: A bar split into thinking and braking parts, and a velocity time graph that falls to zero.

Two forces on one line

Walk a right angle

A bus gains 2 m/s every second

Roll a ball down a slope

A car stops and the passenger does not

Drop a stone into a cup

Acceleration against force for a 2 kg trolley

The same pull on a light box and a heavy box

Time a paper cone as it falls

Two rubber bands pulling

The same change of momentum, a different time

Catch a soft ball two ways

Catch a falling ruler

Key words, A to Z