Question

A ball is launched upward from the ground at an initial vertical speed of v₀ and begins bouncing vertically. Every time it rebounds, it loses a proportion of the magnitude of its velocity due to the inelastic nature of the collision, such that if the speed just before hitting the ground on a bounce is v, then the speed just after the bounce is rv, where r < 1 is a constant. Calculate the total length of time that the ball remains bouncing, assuming that any time associated with the actual contact of the ball with the ground is negligible.

(A) 2v₀/g 1/1 - r

(B) v₀/g r/1 - r

(C) 2v₀/g 1- r/r

(D) 2v₀/g 1/1 - r²

(E) 2v₀/g 1/1 + (1 - r)²

Answer 1

Correct option (A) 2v₀/g 1/1 - r

(advanced question!) The time for one bounce is found from –v = v + (–g)t which gives t = 2v/g. We are summing the time for all bounces, while the velocity (and hence the time) converge in a geometric series with the ratio v_n+1/v_n  = r < 1  to 1/1 - r