Momentum and symmetry

A simple argument for momentum conservation

A billiard ball rolls in, and strikes another identical stationary ball. What happens next? Many billiard players will know. But why does this rather surprising event happen? We suggest you start with a simpler situation and then use your skills at changing your point of view to find a rather simple and satisfying explanation.

Start with two balls, both stationary. They'll both continue to be stationary: there's nothing to disturb the symmetry and no reason for them to do anything else. For the next step, consider two balls moving towards each other with identical velocity. Again you have a symmetric situation, with nothing to break the symmetry, so there are two extremal possibilities: the balls stick together (the balls don't bounce at all), or they bounce back with the same velocity (perfectly bouncy balls). There's a spectrum of possibility between these two, depending on the internal construction of the balls. Let's focus on the perfectly bouncy case and shift our point of view to moving alongside one ball.

That ball is co-moving with us and is therefore at rest. You'll record the other ball as moving in with twice the velocity that it had (as a consequence of your chosen point of view). What will you notice after the collision? The ball that was travelling in towards you now appears at rest, whilst your ball shoots off with the same initial velocity that the other ball had. This is exactly the billiard ball collision.

Taking it further

You can repeat the analysis for the non-bouncy balls, and find that, from your new point of view, they'll move off with half the velocity that the inbound ball had. These results are just the conservation of momentum: the momentum before an interaction is the same as the momentum afterwards, assuming that there are no external forces acting – so nothing to break the symmetry.