Chapter 7 of 8

All 8 chapters
  1. 1Scalars and vectors, readunderstood
  2. 2Velocity and acceleration, readunderstood
  3. 3Newton's first law, readunderstood
  4. 4Newton's second law, readunderstood
  5. 5Weight and falling, readunderstood
  6. 6Newton's third law, readunderstood
  7. 7Momentum, readunderstood
  8. 8Stopping distances, readunderstood

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

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • 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

Two rows of trolleys on a track, with a key saying positive direction. The first row, labelled before, shows a trolley of 1 kg with an arrow to the right of 9 m/s, and a trolley of 2 kg at rest, 0 m/s. The second row, labelled after, shows the two trolleys, 1 kg and 2 kg, joined, with one arrow to the right of 3 m/s.before1 kg2 kg9 m/s0 m/safter1 kg2 kg3 m/spositive direction

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.

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

Answer: 2 m/s to the right

Two rows of trolleys on a track, with a key saying positive direction. The first row, labelled before, shows a trolley of 2 kg with an arrow to the right of 3 m/s, and a trolley of 1 kg at rest, 0 m/s. The second row, labelled after, shows the two trolleys, 2 kg and 1 kg, joined, with one arrow to the right of 2 m/s.before2 kg1 kg3 m/s0 m/safter2 kg1 kg2 m/spositive direction
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.

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

Answer: 2.5 m/s to the right

Two rows of trolleys on a track, with a key saying positive direction. The first row, labelled before, shows a trolley of 3 kg with an arrow to the right of 4 m/s, and a trolley of 1 kg with an arrow to the left, labelled −2 m/s. The second row, labelled after, shows the two trolleys, 3 kg and 1 kg, joined, with one arrow to the right of 2.5 m/s.before3 kg1 kg4 m/s−2 m/safter3 kg1 kg2.5 m/spositive direction
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.

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

Answer: 6 m/s to the right

Two rows of trolleys on a track, with a key saying positive direction. The first row, labelled before, shows a trolley of 2 kg and a trolley of 1 kg, side by side, both 0 m/s. The second row, labelled after, shows the 2 kg trolley with an arrow to the left, labelled −3 m/s, and the 1 kg trolley with a longer arrow to the right of 6 m/s.before2 kg1 kg0 m/s0 m/safter2 kg1 kg−3 m/s6 m/spositive direction
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 bar chart called force (kN). Step 1: one bar for stiff car, 0.025 s, reaching 800. Step 2: two bars, stiff car 0.025 s reaching 800 and crumple zone 0.1 s reaching 200. The scale goes from 0 to 800 in steps of 200.0200400600800stiff carforce (kN)

    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 bar chart called force (kN). Step 1: one bar for stiff car, 0.025 s, reaching 800. Step 2: two bars, stiff car 0.025 s reaching 800 and crumple zone 0.1 s reaching 200. The scale goes from 0 to 800 in steps of 200.0200400600800stiff carcrumple zoneforce (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.

  1. The car goes from 20 m/s to rest, so its change in momentum is the whole of its momentum.
  2. The momentum is 1 000 × 20 = 20 000 kg m/s.
  3. Stiff car: F = 20 000 ÷ 0.025 = 800 000 N.
  4. Crumple zone: F = 20 000 ÷ 0.1 = 200 000 N.
  5. 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.

A bar chart called force (kN). The bar for bare reaches 15. The bar for helmet reaches 3.75. The scale goes from 0 to 15 in steps of 5.051015barehelmetforce (kN)
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.

  1. Stand a short distance from your friend. Ask your friend to toss the sock gently at your hands.
  2. Catch it with stiff, still hands. Notice how it feels.
  3. Catch it again, but let your hands move back with the sock as you catch it. Notice how it feels.
  4. 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 table with the columns Catch and Feels. Stiff hands: sharp push. Hands move back: gentle push.CatchFeelsstiff handssharp pushhands move backgentle push

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.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is the momentum of a car of mass 800 kg moving at 15 m/s?

    Tap your answer.

    1. Not this one. You divided the mass by the velocity. Momentum is the mass times the velocity. Try another.

    2. Not this one. You added the mass and the velocity. Momentum is a product. Try another.

    3. Yes, that is right. p = m × v = 800 kg × 15 m/s = 12 000 kg m/s.

    4. Not this one. That has one zero too many. 800 × 15 = 12 000. Try another.

    See the answer

    12 000 kg m/s

    p = m × v = 800 kg × 15 m/s = 12 000 kg m/s.

    • If you chose A: You divided the mass by the velocity. Momentum is the mass times the velocity.
    • If you chose B: You added the mass and the velocity. Momentum is a product.
    • If you chose D: That has one zero too many. 800 × 15 = 12 000.

  2. Trolley A has a mass of 4 kg and moves right at 2 m/s. It hits trolley B, which also has a mass of 4 kg and is at rest. They stick together. What is their velocity after?

    A row of trolleys on a track, labelled before, with a key saying positive direction. A trolley of 4 kg has an arrow to the right of 2 m/s. A trolley of 4 kg is at rest, 0 m/s.before4 kg4 kg2 m/s0 m/spositive direction
    Tap your answer.

    1. Not this one. That is the velocity of A before. The mass moving is now twice as big, so the velocity is smaller. Try another.

    2. Not this one. The velocity cannot grow in a sticking collision. The total momentum stays the same, over more mass. Try another.

    3. Not this one. You divided the velocity by 4. Divide the total momentum, 8 kg m/s, by the total mass. Try another.

    4. Yes, that is right. Total momentum before = 4 × 2 + 4 × 0 = 8 kg m/s. After, 8 kg move together, so v = 8 ÷ 8 = 1 m/s.

    See the answer

    1 m/s

    Total momentum before = 4 × 2 + 4 × 0 = 8 kg m/s. After, 8 kg move together, so v = 8 ÷ 8 = 1 m/s.

    • If you chose A: That is the velocity of A before. The mass moving is now twice as big, so the velocity is smaller.
    • If you chose B: The velocity cannot grow in a sticking collision. The total momentum stays the same, over more mass.
    • If you chose C: You divided the velocity by 4. Divide the total momentum, 8 kg m/s, by the total mass.

  3. Take right as positive. A ball of mass 5 kg moves left at 3 m/s. What is its momentum?

    Tap your answer.

    1. Not this one. The size is right. The ball moves left, so the momentum is negative. Try another.

    2. Yes, that is right. The velocity is −3 m/s, because it points left. p = m × v = 5 × (−3) = −15 kg m/s.

    3. Not this one. You added the mass and the speed, 5 + 3, then put in the minus sign. Momentum is a product: 5 × (−3) = −15 kg m/s. Try another.

    4. Not this one. You added the mass and the speed. Momentum is a product, and the direction gives a sign. Try another.

    See the answer

    The momentum is −15 kg m/s.

    The velocity is −3 m/s, because it points left. p = m × v = 5 × (−3) = −15 kg m/s.

    • If you chose A: The size is right. The ball moves left, so the momentum is negative.
    • If you chose C: You added the mass and the speed, 5 + 3, then put in the minus sign. Momentum is a product: 5 × (−3) = −15 kg m/s.
    • If you chose D: You added the mass and the speed. Momentum is a product, and the direction gives a sign.

  4. Two trolleys of equal mass move towards each other at the same speed. They stick together. What happens to them?

    Tap your answer.

    1. Not this one. They cannot both keep going. They are stuck together and move as one. Try another.

    2. Not this one. That would be right if one trolley were at rest. Here the two momenta are opposite. Try another.

    3. Yes, that is right. The momenta are +p and −p, so the total before is 0. The total after must also be 0, so the joined trolleys stop.

    See the answer

    They stop.

    The momenta are +p and −p, so the total before is 0. The total after must also be 0, so the joined trolleys stop.

    • If you chose A: They cannot both keep going. They are stuck together and move as one.
    • If you chose B: That would be right if one trolley were at rest. Here the two momenta are opposite.

  5. A cricket fielder pulls her hands back as she catches a fast ball. Why does that make the catch hurt less?

    Tap your answer.

    1. Yes, that is right. The ball loses the same momentum. A longer time makes F = Δp ÷ t smaller.

    2. Not this one. The ball loses all its momentum either way. It is the time to stop that changes. Try another.

    3. Not this one. Mass does not change. The time of the stop is what changes. Try another.

    See the answer

    The ball takes longer to stop, so the force on her hands is smaller.

    The ball loses the same momentum. A longer time makes F = Δp ÷ t smaller.

    • If you chose B: The ball loses all its momentum either way. It is the time to stop that changes.
    • If you chose C: Mass does not change. The time of the stop is what changes.

  6. A crumple zone makes the stop of a car last longer. Compared with a stiff car stopping from the same velocity, the change of momentum of the passengers is...

    Tap your answer.

    1. Not this one. It is not bigger. They go from the same velocity to rest. Try another.

    2. Not this one. It is not smaller. They still go from the same velocity to rest. Try another.

    3. Not this one. They do stop, so their momentum changes by all of it. Try another.

    4. Yes, that is right. The passengers go from the same velocity to rest, so the change of momentum is the same. The longer time gives a smaller force, F = Δp ÷ t.

    See the answer

    the same, but the force is smaller.

    The passengers go from the same velocity to rest, so the change of momentum is the same. The longer time gives a smaller force, F = Δp ÷ t.

    • If you chose A: It is not bigger. They go from the same velocity to rest.
    • If you chose B: It is not smaller. They still go from the same velocity to rest.
    • If you chose C: They do stop, so their momentum changes by all of it.

  7. From an earlier chapter

    Which pair of forces is a third law pair?

    Tap your answer.

    1. Yes, that is right. They are two forces of the same type, gravity, on two objects, with the same size and opposite directions.

    2. Not this one. Both forces act on the book. They are balanced forces, not a pair. Go to S4.8.6. Try another.

    3. Not this one. They are different types of force, and they are not partners. Go to S4.8.6. Try another.

    See the answer

    The Earth pulls the Moon, and the Moon pulls the Earth.

    They are two forces of the same type, gravity, on two objects, with the same size and opposite directions.

    • If you chose B: Both forces act on the book. They are balanced forces, not a pair. Go to S4.8.6.
    • If you chose C: They are different types of force, and they are not partners. Go to S4.8.6.

  8. A cart of mass 60 kg moves at 3 m/s. What is its momentum?

    See the answer

    180 kg m/s

    p = m × v = 60 kg × 3 m/s = 180 kg m/s.

  9. Trolley A has a mass of 3 kg and moves at 5 m/s. It hits trolley B, of mass 7 kg, at rest. They stick together. What is their velocity after?

    See the answer

    1.5 m/s

    The total momentum is 3 × 5 + 7 × 0 = 15 kg m/s. After, 10 kg move together, so v = 15 ÷ 10 = 1.5 m/s.

  10. A ball of mass 0.5 kg moving at 8 m/s is stopped by a wall in 0.1 s. What is the average force on the ball?

    See the answer

    40 N

    The change of momentum is 0.5 × 8 = 4 kg m/s. F = Δp ÷ t = 4 ÷ 0.1 = 40 N.

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.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (825 KB). It opens on any phone.