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
`(1)/(2) mv^2`. Therefore, the total kinetic energy before the collision is

`K_i = (1)/(2) mv^2 + (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 = (1)/(2) mv_1^2 + (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.