Question

Do conservation of momentum and conservation of energy lead to the same results ?

Answer 1

A simple way to see this is to try and calculate the final velocities of two particles after an elastic collision. Pick an arbitrary starting configuration (such that they will interact), and calculate their initial kinetic energy, K. We know that when we repeat this calculation at some time after the collision, we need to get the same value, so we set KK =12m1v21+12m2v22=12m1v12+12m2v22. Now we are solving for the two speeds but we only have one equation. This means that in general there are multiple speed pairs the particles can have and conserve energy. It’s even worse though, we have no idea which direction the particles will go. The conservation of energy alone is not sufficient. We need to introduce the conservation of momentum as a separate assumption to uniquely identify a solution that gives us the magnitude AND direction of the particles velocity.

On a deeper level, conservation of momentum and conservation of energy come from two separate symmetries about the laws of physics. Conservation of energy is due to the fact that physical laws are time invariant, in other words the physics of tomorrow is the same as today, and is the same as it was 100 years ago. Conservation of momentum comes from the fact that physical laws are translationally invariant, in other words the fundamental physics where I am is the same as where you are.