Question

A ball is held at rest at some height above a hard, horizontal surface. Once the ball is released it falls, hits the surface, and starts bouncing vertically up and down. Suppose that with each bounce the ball loses a fixed fraction p (with 1>p>0) of its energy. This loss could be due to a number of reasons (inelasticity, drag, etc) that are left unspecified.

1. How many times will the ball bounce before coming to rest (if at all)? Provide a detailed explanation of your reasoning, not simply a one-line answer.
2. How long will it take for the ball to come to rest (if at all)? Give your answer as a formula that contains as variables only p and the time T₁ from the moment that the ball was released to the first contact with the horizontal surface.

Answer 1

initial height dropped =h = gT₁²/2

T₁ - time from the moment that the ball was released to the first contact with the horizontal surface.

initial energy T = mgh

after first bounce energy loss = mghp

height it will raise after first bounce mgh₁ = mgh(1-p)

after each bounce its energy is reduced by p

after n bounces it will raise to a height

mgh_n = mgh(1-p)ⁿ

h_n = h(1-p)ⁿ = gT₁²/2 *(1-p)ⁿ

  

theoritically h_n can never become 0. It can be closer to 0 or can be negligibly small after n bounces, depending on the actual values of p and T₁ .