Question

Question #793fe

Answer 1

The ratio will be `(1-e)/(1+e)`.

Explanation:

By definition, the coefficient of restitution represents the ratio between the relative velocity of separation and the relative velocity of approach.

Simply put, the coefficient of restitution tells how much kinetic energy is preserved after a collision.

Let's say that, after impact, the velocity of sphere `A` will be `v_A` and the velocity of sphere `B` will be `v_B`.

The Law of conservation of linear momentum tells you that the overall momentum before the collision must be equal to the overall momentum after a collision.

Since the spheres have identical masses, you can write

`m * u + underbrace(m * 0)_(color(blue)("sphere B is stationary")) = m * v_A + m * v_B`

This is of course equivalent to

`cancel(m) * u + cancel(m) * 0 = cancel(m) * v_A + cancel(m) * v_B`

`u = v_A + v_B`

From the definition of the coefficient of restitution, you know that

`e = (v_B - v_A)/(u - 0) = (v_B - v_A)/u`

Replace `u` from the first equation into this one to get

`e = (v_B - v_A)/(v_A + v_B) <=> e * (v_A + v_B) = v_B - v_A`

`e * v_A + e * v_B = v_B - v_A => v_A * (1 + e) = v_B * (1-e)`

This is equivalent to

`v_A/v_B = color(green)((1-e)/(1 + e))`