Momentum Matters

Momentum Matters

A rolling marble is easy to stop and a rolling lorry is not, and the difference is not simply that one is heavier. A lorry crawling at walking pace is also easy to stop. What matters is mass and velocity together, and that combination has a name, a symbol and one extraordinary property: in a closed system it is never created or destroyed, whatever happens inside. That single fact lets you work out what comes out of a collision without knowing anything at all about the forces inside it. (Higher tier.)

The words and symbols

Five terms, and the third is the one that catches people out:

Find the momentum

A lorry of mass 1000 kg travels at 20 m/s. What is its momentum, in kg m/s?

Two trolleys, one track

Two identical 2 kg trolleys approach each other head-on along the same track, each travelling at 3 m/s. What is the total momentum of the system before they meet?

• Zero. Momentum is a vector, so one trolley has +6 kg m/s and the other −6 kg m/s, and they cancel exactly
• 12 kg m/s, since each carries 6 and there are two of them
• 6 kg m/s, because the two are identical so one effectively counts
• It cannot be found without knowing which trolley is heavier

Two things that might be conserved

In every collision, momentum is conserved. Kinetic energy usually is not, and knowing which is which is worth a mark on its own:

The missing energy

Two trolleys collide and stick together. Calculating carefully, the total momentum afterwards is exactly what it was before, but the total kinetic energy afterwards is only two thirds of what it was. Has conservation of momentum been broken, and where has the energy gone?

• Momentum is fine: it is conserved and the calculation confirms it. The missing kinetic energy was transferred to heat, sound and the work done permanently deforming the trolleys as they locked together. Energy is conserved too, but not in its kinetic form
• Yes, conservation has been broken. If energy was lost then momentum must have been lost with it
• There must be an arithmetic error, since both quantities have to be conserved in every collision
• The kinetic energy was converted into extra momentum, which is why momentum stayed the same

A sticky collision

A 2 kg trolley moving at 3 m/s collides with a stationary 1 kg trolley, and the two stick together. What is their velocity immediately afterwards, in m/s?

An explosion, worked

A 5 kg rifle fires a 0.05 kg bullet at 400 m/s. Find the recoil velocity of the rifle. Before. Nothing is moving, so the total momentum is zero. That is the whole basis of the calculation, and it is why explosions are often easier than collisions. After. The total must still be zero, so the two momenta must cancel: bullet momentum + rifle momentum = 0. Taking forwards as positive, the bullet carries 0.05 × 400 = +20 kg m/s. The rifle must therefore carry −20 kg m/s, and with a mass of 5 kg its velocity is −20 ÷ 5 = −4 m/s. The minus sign is the answer, not an error: it says the rifle moves backwards, at 4 m/s. Always state a direction with a recoil velocity, because the sign is carrying real information.

Pushing off

Two ice skaters stand still, face to face, and push apart. The 75 kg skater moves off at 2 m/s. How fast does the 50 kg skater move, in m/s?

You cannot change Δp, only Δt

A resultant force changes momentum, and the relationship is: F = m Δv ÷ Δt Now read it as a piece of engineering advice. In a crash, the change in momentum is already settled: you were doing 30 mph and you are about to be doing nothing, and no device in the car can alter either of those numbers. Δp is fixed before the safety features get a say. So the only quantity left that anything can influence is Δt. Every safety feature ever fitted to a vehicle is an attempt to make the same momentum change take longer, because a longer time is the only route to a smaller force. This is why "the airbag absorbs your momentum" is wrong and worth un-learning. Your momentum change is identical with or without it. What differs is whether it happens in five milliseconds against a steering column or in fifty against a cushion, and that factor of ten is the whole benefit.

In the crash

A 1000 kg car travelling at 20 m/s is brought to rest in a collision. Three questions from the design office.

• Without a crumple zone the car stops in 0.4 s; with one it stops in 4 s. What average force acts on the car in the second case?
• A colleague says the crumple zone "reduces the momentum change on the passengers". Is that right?
• If a longer stopping time is always better, why not build the entire car as one enormous crumple zone?

What does each one actually do?

How a crumple zone protects you

Put the reasoning into order, starting from what cannot be changed.

• The passenger is travelling at 20 m/s and must end at rest, so the change in momentum is fixed before anything else is decided
• The crumple zone folds progressively rather than the car stopping dead
• The same change in momentum is therefore spread over a longer time
• Since force is the momentum change divided by the time, a longer time gives a smaller average force
• A smaller force on the body means less serious injury

Momentum summary

Momentum is mass times velocity, and because it is a _____ it has direction, so opposing motions cancel. In a closed system the total is always conserved, which is what makes a collision solvable without knowing anything about the forces inside it. Kinetic energy, by contrast, is conserved only in an _____ collision; otherwise some becomes heat, sound and deformation. In a crash the change of momentum is _____ by the speed you were doing, so the only quantity a safety feature can alter is the _____ the change takes, which is how a smaller force is achieved.