You already have one way to describe how hard something is to stop. Kinetic energy, from Unit 3, is a scalar and it grows as the square of the speed. Momentum is the other way, it is a vector, and it grows as the first power. A loaded truck rolling slowly and a bullet are both hard to stop, and they are hard to stop differently.

There is a second reason this quantity is about to matter more than kinetic energy did. In the next topic you will meet it inside a derivative, and the law you have been calling Newton’s second law will turn out to be a special case of a statement about momentum. Keep that in view while you read this page.

1. Momentum is mass times velocity

\(\vec{p} = m\vec{v}\)kilogram meters per second

There is no separate name for the unit. It is written \(\mathrm{kg\cdot m/s}\) and read as it is written. If a quantity you have calculated does not come out in kilogram meters per second, it is not a momentum.

Momentum is a vector. It points in exactly the same direction as the velocity, because mass is a positive scalar and multiplying a vector by a positive scalar does not turn it. On a straight track a momentum to the right is positive and a momentum to the left is negative, and that sign is part of the answer rather than decoration on it.

Because the relationship is linear in both inputs, scaling questions are easy and the exam asks them often. Double the mass and you double the momentum. Triple the speed and you triple it. Do both and you multiply by six. Nothing is squared here, which is exactly where momentum parts company with kinetic energy.

2. How momentum and kinetic energy scale differently

Momentum answers a question speed alone cannot: how much motion there is to get rid of. A loaded shopping cart and an empty one rolling at the same speed are not equally easy to stop, and the arithmetic says so before your arms do.

\(p = mv \qquad K = \tfrac{1}{2}mv^{2} = \frac{p^{2}}{2m}\)the same motion, two descriptions

The second form of the kinetic energy is worth having. It says that at a fixed momentum, a heavier object carries less kinetic energy. That is why a heavy slow object and a light fast one with the same momentum are not equally dangerous, and it will come back in Topic 4.4.

3. A system has one momentum, and it is a sum

Pick any group of objects and call it a system. The momentum of that system is the vector sum of the momenta of its parts. That is all, and it is why the bottom arrow in the first simulation is simply the two arrows above it laid head to tail.

\(\vec{p}_{\text{sys}} = \sum_i \vec{p}_i = \sum_i m_i \vec{v}_i\)add the vectors, not the sizes

Adding the sizes is the most common way to get this wrong. Two carts of momentum \(6\ \mathrm{kg\cdot m/s}\) moving toward each other have a total of zero, not twelve. The sizes do add to twelve, and the sizes are not what the physics asked for.

For a system whose mass is spread out rather than concentrated in a few objects, the sum becomes an integral, \(\vec{p} = \int \vec{v}\,dm\). You will not need to evaluate one in this unit, but the idea is the same and the notation should not surprise you when it appears.

4. Two models: the collision and the explosion

Momentum earns its keep in two situations the course framework names as models.

A collision is an interaction in which the forces the objects exert on each other are much larger than any net external force on them. For the fraction of a second two carts are touching, the push between them dwarfs friction from the track and everything else acting from outside. That is what lets you ignore the outside world for the length of the interaction.

Because only the state before and the state after are examined, each object can be treated as a single point carrying a mass. Its shape, its spin and where it was struck never enter the analysis. The framework calls that the object model, and it is allowed here precisely because the inside of the interaction is never looked at.

An explosion is the same idea pointed the other way. Forces inside the system move its parts apart: a spring released between two carts, a firework, a person stepping off a skateboard. Nothing is added from outside, so whatever momentum the system had before, it still has after.

Notice what the second simulation refuses to do. It never lets the two arrows come out unequal. Change the split, change how hard the spring pushes, and the two momenta still match in size and oppose in direction. That is not the simulation being polite. It is the only outcome the physics allows, and the next two topics are about why.