Two balls with equal masses, m, and equal speed, v, engage in a head on elastic collision. what is the final velocity of each ball, in terms of m and v

The collision is elastic. This means that both momentum and kinetic energy are conserved after the collision.

- Let's start with conservation of momentum. The initial momentum of the total system is the sum of the momenta of the two balls, but we should put a negative sign in front of the velocity of the second ball, because it travels in the opposite direction of ball 1. So ball 1 has mass m and speed v, while ball 2 has mass m and speed -v:

So, the final momentum must be zero as well:

Calling v1 and v2 the velocities of the two balls after the collision, the final momentum can be written as

From which


- So now let's apply conservation of kinetic energy. The kinetic energy of each ball is . Therefore, the total kinetic energy before the collision is

the kinetic energy after the collision must be conserved, and therefore must be equal to this value:
(1)
But the final kinetic energy, Kf, is also

Substituting  as we found in the conservation of momentum, this becomes

we also said that Kf must be equal to the initial kinetic energy (1), therefore we can write 


Therefore, the two final speeds of the balls are



This means that after the collision, the two balls have same velocity v, but they go in the opposite direction with respect to their original direction.