Question

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

Answer 1

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:
$p_i = p_1-p_2 = mv-mv =0$
So, the final momentum must be zero as well:
$p_f = 0$
Calling v1 and v2 the velocities of the two balls after the collision, the final momentum can be written as
$p_f = mv_1 + mv_2 = 0$
From which
$v_1 = -v_2$

- So now let's apply conservation of kinetic energy. The kinetic energy of each ball is $\frac{1}{2} mv^2$. Therefore, the total kinetic energy before the collision is
$K_i = \frac{1}{2} mv^2 + \frac{1}{2} mv^2 = mv^2$
the kinetic energy after the collision must be conserved, and therefore must be equal to this value:
$K_f = K_i = mv^2$ (1)
But the final kinetic energy, Kf, is also
$K_f = \frac{1}{2} mv_1^2 + \frac{1}{2}mv_2^2$
Substituting $v_1 = -v_2$ as we found in the conservation of momentum, this becomes
$K_f = mv_2 ^2$
we also said that Kf must be equal to the initial kinetic energy (1), therefore we can write 
$mv_2^2 = mv^2$

Therefore, the two final speeds of the balls are
$v_2 = v$
$v_1 = -v_2 = -v$

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.