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:
[tex]p_i = p_1-p_2 = mv-mv =0[/tex]
So, the final momentum must be zero as well:
[tex]p_f = 0[/tex]
Calling v1 and v2 the velocities of the two balls after the collision, the final momentum can be written as
[tex]p_f = mv_1 + mv_2 = 0[/tex]
From which
[tex]v_1 = -v_2[/tex]
- So now let's apply conservation of kinetic energy. The kinetic energy of each ball is [tex] \frac{1}{2} mv^2[/tex]. Therefore, the total kinetic energy before the collision is
[tex]K_i = \frac{1}{2} mv^2 + \frac{1}{2} mv^2 = mv^2 [/tex]
the kinetic energy after the collision must be conserved, and therefore must be equal to this value:
[tex]K_f = K_i = mv^2[/tex] (1)
But the final kinetic energy, Kf, is also
[tex]K_f = \frac{1}{2} mv_1^2 + \frac{1}{2}mv_2^2 [/tex]
Substituting [tex]v_1 = -v_2[/tex] as we found in the conservation of momentum, this becomes
[tex]K_f = mv_2 ^2[/tex]
we also said that Kf must be equal to the initial kinetic energy (1), therefore we can write
[tex]mv_2^2 = mv^2[/tex]
Therefore, the two final speeds of the balls are
[tex]v_2 = v[/tex]
[tex]v_1 = -v_2 = -v[/tex]
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.
