How Can a Collision Conserve Momentum, But Not Kinetic Energy?

When two objects interact in a collision, momentum is conserved where momentum (p) is defined as the product of mass (m) and velocity (v). However, if the kinetic energy also depends on mass and velocity (1/2 mv²) then shouldn’t it also be conserved? Nope. Let’s work through the whole thing.

Simple Collision Between Two Point Particles

Let’s start with the simplest example possible (maybe). Imagine that there is a particle with mass m moving in the x-direction with a velocity v (called particle A). It collides with another particle with mass m, but that’s stationary (particle B). Oh, it’s a head on collision so that every stays in 1 dimension (the x-direction).

The key thing idea is that during the collision object A pushes on B and B pushes back on A. But these are the SAME interaction such that the magnitude of A on B is equal to B on A. Now let’s consider the momentum principle. It says the following:

We can do the same thing for particle B, but remember the two forces are equal and opposite. This means that the change in momentum for A must be the opposite of mass B (assuming the time intervals are the same — but why wouldn’t they be).

Boom. That’s conservation of momentum. Can we use this to find the final velocity of both particles after the collision? Actually, no. Suppose the final velocity of A is v_A and for B it’s v_B. Setting up conservation of momentum gives the following (in just the x-direction so we don’t need the vectors):