Momentum and Collisions

Momentum measures how hard something is to stop. It combines the two properties that decide that difficulty — how much mass is moving and how fast — and it has a useful property in collisions: the total is unchanged, which lets you work out what happens after an impact from what was happening before.

Momentum

Momentum is mass multiplied by velocity.

The unit has no special name; it is simply kilogram metres per second. Because velocity is a vector, momentum is too, so a direction belongs with it and the sign matters when objects move opposite ways. A loaded lorry at walking pace and a small car at speed can carry the same momentum, which matches the everyday sense that both would be equally awkward to bring to a halt.

It is worth separating momentum from two things it is easily confused with. It is not force — force is what changes momentum. And it is not kinetic energy, which the next module covers, even though both involve mass and speed: momentum uses the velocity once and keeps its direction, while kinetic energy uses the speed squared and has no direction at all.

Momentum is conserved

In any collision or explosion where no outside force interferes, the total momentum of everything involved is the same afterwards as it was before. This is genuinely useful: you rarely know the details of what happens during an impact, and you do not need to. Add up the momentum before, add up the momentum after, and set them equal.

Why a longer impact is gentler

Bringing something to a stop means removing all its momentum, and how much force that takes depends on how long you allow. Stop a moving mass in a hundredth of a second and the force is enormous; stop the same mass over half a second and the same momentum change is spread out, so the force is far smaller. This follows directly from Newton's second law — force is what changes momentum, so a given change delivered slowly needs less of it.

The numbers make the point better than words. Removing 60 kg·m/s of momentum in a hundredth of a second needs $\frac{60}{0.01} = 6{,}000$ newtons; removing the same 60 kg·m/s over half a second needs $\frac{60}{0.5} = 120$ newtons. Same momentum change, fifty times less force — bought entirely with time.

Every piece of impact protection is built on this. A car's crumple zone extends the collision from milliseconds to a substantial fraction of a second, which is why a car that visibly destroys itself protects its occupants better than a rigid one that does not. An airbag, a crash mat, a boxer riding a punch and bending your knees on landing all do the identical thing: they lengthen the time, so they lower the force.

Putting it to work

Explosions and recoil

Conservation works just as well when things fly apart as when they collide. Before a firework bursts or a rifle is fired, the total momentum is zero — nothing is moving. Afterwards the pieces are moving, and the total must still be zero, which is only possible if the momenta cancel: whatever goes one way is exactly matched by momentum going the other. That is recoil. A light bullet leaving fast and a heavy rifle moving back slowly carry equal and opposite momenta, and because the rifle's mass is so much larger its velocity is correspondingly smaller.

Sticking together or bouncing apart

Collisions come in two broad kinds, and momentum is conserved in both. Sometimes the objects stick together and move off as one, as in the worked example above — the wreckage of two cars, or two trolleys with putty between them. Sometimes they bounce apart and separate at different speeds, as billiard balls do. The total momentum before and after is identical either way, so the same equation solves both; the only difference is whether the objects share one velocity afterwards or have two. When a question describes objects that "move off together", it is telling you they now have a single combined mass and a single velocity, which simplifies the arithmetic considerably.

Finally, a word on when conservation applies. The rule holds when no outside force acts on the system you are considering, which in practice means the collision is quick enough that friction and other external effects have not had time to matter. Over a longer period momentum clearly is not conserved — a rolling ball slows and stops, losing all of it to friction with the ground. So the technique works for the instant of impact and its immediate aftermath, which is exactly the window questions ask about. If a problem describes something coming to rest over a long distance, that is a friction problem rather than a conservation one.

Why it matters in aviation

Momentum is the cleanest way to describe how an engine produces thrust. Air enters at the speed of the aircraft and leaves faster, so the engine is adding momentum to that stream continuously, and the reaction to that rate of change of momentum is the forward force on the airframe. Because the rate is a mass of air per second multiplied by the change in its velocity, the same thrust can be had by moving a great deal of air a little faster or a little air much faster — which is the design difference between a large-diameter fan and a pure jet, not a difference in the physics.

The impulse form of the same idea sets the loads a structure has to survive. Landing gear exists to remove the aircraft's downward momentum, and since the force depends on how long that takes, the shock strut and the tyre are there to spread the stop over a greater distance and a longer time, reducing the peak force in exactly the way a crumple zone does. A bird strike is the opposite case: a small mass arriving at flight speed transfers its momentum in a very short time, so the force is large, and that is why engines and windshields are certified against such an impact rather than merely built stiff.

What the test asks

The first trap is forgetting that momentum has a direction: in a head-on collision the two momenta have opposite signs and partly cancel, and treating both as positive gives a total that is far too large. The second is conflating momentum with kinetic energy — in the collision above the momentum is unchanged while the kinetic energy is not, and an option will invite you to say both survive. The third is the crumple-zone question, where the tempting wrong answer is that a stiffer car is safer.