Momentum

What this lesson covers in the specification

Incandio is aligned to this specification. It is not published by, endorsed by or affiliated with Pearson, and it reproduces none of Pearson's wording — the statement numbers are given so you can check every lesson against your own copy.

• 1.2 — Use the units newton metre and kilogram metre/second (bold P statement — Paper 2 only)
• 1.25 — Use p = m × v (bold P statement — Paper 2 only)
• 1.26 — Use momentum to explain safety features (bold P statement — Paper 2 only)

Specification section 1(a) Units — also taught in Speed, and What a Distance–Time Graph Is Telling You.

1 · Understand it

No exam language yet. The only question this section answers is: do I actually understand what is happening?

Ask which is harder to stop: a bicycle at 10 m/s or a lorry crawling at 1 m/s. The bicycle is faster, the lorry is heavier, and neither fact settles it on its own. What settles it is the two multiplied together — and that product is important enough to have a name. MOMENTUM is mass × velocity, and it measures how much motion something has in the sense that matters when you try to change it.

Newton did not call it momentum. In the Principia he called it the quantity of motion, and he defined it before he stated any of his laws, because his second law is really a statement about how momentum changes. The bicycle at 10 m/s with a mass of 15 kg has 150 kg m/s. A 4000 kg lorry at 1 m/s has 4000 kg m/s. The lorry is harder to stop by a factor of nearly thirty, and it is barely moving.

Two things about the unit are worth pausing on, because the specification names units explicitly. Momentum is measured in KILOGRAM METRES PER SECOND, kg m/s, and there is no special name for it — you simply carry the units of mass and velocity through the multiplication, which is a useful habit for checking any answer. The other unit named in the same statement is the NEWTON METRE, N m, which is a force multiplied by a distance. That is the unit of a moment, the turning effect of a force, and it belongs to the next page but the same rule produces it: multiply the quantities, multiply the units.

Momentum is a VECTOR, because velocity is a vector and mass is just a number. In a straight-line problem this means one direction has to be chosen as positive and everything else follows from that choice. A 2 kg ball moving right at 3 m/s has momentum +6 kg m/s; the same ball moving left at 3 m/s has −6 kg m/s. Those are not the same quantity with a cosmetic sign — they are opposite, and the difference between them is 12 kg m/s, not zero, which is exactly why a ball that bounces back off a wall has a much larger change in momentum than one that stops dead.

Momentum, and the change in momentum, for a ball that bounces

A 0.20 kg ball travelling right at 8.0 m/s hits a wall and rebounds at 6.0 m/s. Find its momentum before, its momentum after, and the change in momentum.

1. Take rightwards as positive.
2. Before: p = m × v = 0.20 × 8.0 = +1.6 kg m/s.
3. After: it is moving LEFT, so v = −6.0 m/s and p = 0.20 × (−6.0) = −1.2 kg m/s.
4. Change in momentum = final − initial = (−1.2) − (+1.6) = −2.8 kg m/s.
5. The magnitude is 2.8 kg m/s, and the minus sign says the change is directed to the left — which is the direction of the force the wall applied.

Answer: +1.6 kg m/s before, −1.2 kg m/s after, a change of 2.8 kg m/s directed leftwards.

Notice what would have happened if the signs had been ignored: 1.6 − 1.2 = 0.4 kg m/s, which is seven times too small. The rebound is the case that catches people out, and it is worth remembering the general result — a ball that bounces back undergoes a bigger change in momentum than one that stops, so the wall must push on it harder.

Now the part of this page that matters outside an examination hall. In a crash, a passenger has a certain momentum before and zero momentum afterwards. That change is FIXED: it is decided by their mass and by how fast the car was going, and no engineering can alter it. What engineering can alter is the TIME over which the change happens. Spread the same change of momentum over a longer time and the force needed to produce it is smaller — in fact the force is inversely proportional to the time, so five times the time means one fifth of the force. And the force is what does the injuring.

Think of it like catching a cricket ball

Anyone who has caught a hard ball knows to pull their hands back as the ball arrives. The ball's momentum has to be reduced to zero either way — that is not negotiable, and moving your hands does not change it by a single unit. What moving your hands changes is how long the reduction takes. Catch it with rigid hands and the change happens in a few thousandths of a second, and the force is large enough to hurt. Draw your hands back and the same change is spread over perhaps a tenth of a second, and the force is a fraction of the size. Every safety feature in a car is a way of pulling your hands back on your behalf.

The safety features, and the identical argument behind each

1. A CRUMPLE ZONE at the front of a car is designed to fold and deform rather than stay rigid. The car takes longer to come to rest, so the passenger's momentum changes over a longer time and the force on them is smaller.
2. A SEAT BELT is slightly stretchy and is designed to extend a little under load. The passenger is brought to rest over a longer time and distance than a rigid restraint would allow — and it also stops them continuing forwards into the windscreen.
3. An AIR BAG inflates and then deflates through vents as the passenger presses into it, so they decelerate over a longer time; it also spreads the force over a larger area of the body.
4. A CRASH HELMET has a crushable liner that collapses on impact, extending the time over which the head is brought to rest.
5. A CUSHIONED PLAYGROUND SURFACE does the same for a falling child: the surface compresses, so the child stops over a longer time than they would on concrete.
6. In every case the change of momentum is exactly the same as it would have been. Only the time is different, and therefore only the force is different.

That last sentence is the whole of statement 1.26, and it is the sentence that has to appear in an answer. A response saying an airbag 'absorbs the impact' or 'cushions the blow' describes the sensation rather than the physics and earns very little. The examinable version names the change in momentum, states that it is unchanged, says the time is increased, and concludes that the force is therefore reduced.

2 · Grade 9 Notes

A different job from the section above. You have already understood it; this is the precise set of things to LEARN — definitions to reproduce word for word, processes in order, equations with units, and the answers that score full marks.

p = m × v

p is momentum in kilogram metres per second (kg m/s), m is mass in kilograms (kg), v is velocity in metres per second (m/s)

Units: kg m/s. Rearranged: m = p ÷ v and v = p ÷ m. Momentum is a VECTOR — it takes the direction of the velocity.

Learn this definition · Momentum

The product of an object's mass and its velocity. It is a vector quantity, measured in kilogram metres per second (kg m/s), and it points in the direction of the velocity.

Statement 1.2 — the two units named, and where each comes from

Kilogram metre per second (kg m/s)

The unit of momentum: kilograms multiplied by metres per second. It has no special name.

Newton metre (N m)

The unit of a moment: newtons multiplied by metres, from force × perpendicular distance from the pivot.

The rule behind both

Multiply the quantities and multiply their units — checking units this way catches a wrongly rearranged equation.

Finding a change in momentum when the direction reverses

1. Choose one direction as positive and state your choice.
2. Write the momentum before, using a negative velocity if it is travelling the other way.
3. Write the momentum after, applying the same sign rule.
4. Change in momentum = final momentum − initial momentum.
5. The sign of the answer gives the direction of the change, and therefore of the force that caused it.

Stopping dead against bouncing back — why the second is worse

A. STOPS DEAD: the momentum goes from mu to zero, so the change is mu. Only the original motion has to be removed.

B. BOUNCES BACK: the momentum goes from +mu to −mv, so the change is mu + mv. The motion is removed AND reversed, so the force is larger.

Statement 1.26 — the four-step answer that gets full marks for any safety feature

• The passenger's CHANGE IN MOMENTUM is fixed by their mass and the speed of the car — the device cannot alter it
• The device INCREASES THE TIME over which that change happens, by crumpling, stretching, inflating or compressing
• The same change of momentum spread over a longer time requires a SMALLER FORCE
• A smaller force on the passenger means less injury — and some devices also spread the force over a LARGER AREA

Model answer [4 marks]

Explain, in terms of momentum, how a crumple zone reduces injuries in a collision. [4]

In the collision the passenger's momentum is reduced to zero, and that change in momentum is fixed by their mass and the speed the car was travelling at. The crumple zone is designed to deform rather than remain rigid, so the car and the passenger are brought to rest over a longer time than they would be otherwise. The same change of momentum taking place over a longer time means a smaller rate of change of momentum, and therefore a smaller force acts on the passenger. It is the force that causes injury, so a smaller force means less serious injuries.

Model answer [3 marks]

A 0.15 kg ball moving at 12 m/s to the right rebounds from a wall at 9.0 m/s to the left. Calculate the magnitude of the change in momentum. [3]

Taking rightwards as positive, the momentum before is 0.15 × 12 = +1.8 kg m/s. After the bounce the ball moves left, so its velocity is −9.0 m/s and its momentum is 0.15 × (−9.0) = −1.35 kg m/s. The change is the final momentum minus the initial momentum: (−1.35) − (+1.8) = −3.15 kg m/s, so the magnitude of the change is 3.15 kg m/s, directed to the left.

Not this: An air bag works by absorbing the force of the impact.

This: It changes how long the passenger takes to stop. The change in momentum is the same with or without it; because that change is spread over a longer time, the force needed to produce it is smaller. 'Absorbs the force' names no mechanism and earns no marks.

Mark-losing trap. Momentum is a VECTOR. A rebound needs a negative velocity, or the change comes out far too small.

Mark-losing trap. Change in momentum = FINAL − INITIAL, in that order. The sign tells you the direction of the force.

Mark-losing trap. Safety answers must say the change in momentum is UNCHANGED and only the time is longer.

Mark-losing trap. kg m/s has no special name. Write the units out; they are worth a mark.

3 · Prove it — the five questions

The five questions climb Grade 6 → Grade 7 → Grade 8 → Grade 9 → Grade 9 challenge, and are marked inside Incandio on your own device, by rule, with an authored diagnosis of the mistake you actually made. The mark schemes stay in the app so that the practice is worth doing; the questions themselves are here.

1. Grade 6 · Calculate [2 marks] — A trolley of mass 3.0 kg moves at 4.0 m/s. Calculate its momentum.
2. Grade 7 · State [2 marks] — Which of these correctly describes what a car's crumple zone does in a collision?
3. Grade 8 · Calculate [3 marks] — A 0.40 kg ball travelling right at 5.0 m/s bounces off a wall and travels left at 3.0 m/s. Calculate the magnitude of its change in momentum.
4. Grade 9 · Explain [5 marks] — A cyclist wearing a helmet falls and strikes their head on the road. Select every statement that belongs in a full-mark explanation of how the helmet reduces the injury.
5. 9+ · Analyse [6 marks] — Two cars of equal mass hit a wall at the same speed. Car A crumples and stops in 0.15 s. Car B is completely rigid and stops in 0.030 s. Select every statement that belongs in a full-mark analysis of the forces on the drivers.

The people behind this science

Two ways into the same idea — the one who defined it before he wrote any of his laws, and the one who used it to put people somewhere exact and bring them home. Inside Incandio each of them answers knowing exactly which lesson you have just finished.

Isaac Newton — the one who defined it before he wrote any of his laws

Momentum is Newton's own quantity, and its position in the Principia tells you how central he thought it was: he defines the quantity of motion — mass multiplied by velocity — in the definitions at the very start, before the three laws, because the second law is a statement about how that quantity changes. He worked with it partly because collisions were the problem of the age; the Royal Society had set the question of what is conserved when two bodies strike, and the answers submitted shaped what he wrote.

• “Why did you define the quantity of motion before stating your laws?”
• “What made mass times velocity the right quantity to track?”
• “How is your second law really a statement about momentum?”
• “What happens to the motion of two bodies that strike each other?”
• “Is a slow heavy body genuinely harder to stop than a fast light one?”

Katherine Johnson — the one who used it to put people somewhere exact and bring them home

Spaceflight is momentum arithmetic and nothing else: a rocket gains momentum only by throwing mass backwards, a capsule changes orbit only by changing its momentum, and a re-entry is the controlled destruction of an enormous quantity of it. Johnson calculated those trajectories by hand at NASA, and when the electronic computers produced the numbers for John Glenn's flight he asked for her to check them before he would fly. She is the person on this page who knows what it means for the calculation to be the safety feature.

• “How does a rocket change its motion with nothing to push against?”
• “What did you actually calculate for John Glenn's flight?”
• “How do you check a machine's arithmetic by hand?”
• “What has to be got right to bring a capsule back to a chosen place?”
• “How much of spaceflight comes down to mass multiplied by velocity?”

Carry on through the course

• Previous lesson: Hooke's Law and Elasticity
• Next lesson: Conservation of Momentum
• Speed, and What a Distance–Time Graph Is Telling You — Speed as a rate, the units physics insists on, and how to read a journey straight off a graph.
• Acceleration and the Velocity–Time Graph — What acceleration actually measures, why its unit looks strange, and how one graph gives you both acceleration and distance.
• Forces, Vectors and Scalars — What a force actually does, why some quantities need a direction to mean anything, and why that makes a force one of them.
• Resultant Force, Friction and F = ma — Why a moving object stops, what really happens when you add forces up, and the one equation that connects force to motion.
• Weight and Terminal Velocity — What weight actually is, why it is not the same as mass, and what happens to a falling object once the air starts pushing back.
• All of Physics · Physics key terms · required practicals · equations · Incandio Science · The 4PH1 specification