Question

Bouncy ball: A ball is dropped from a height of H meters. Under ideal conditions (vacuum), after each bounce it returns to 1/3 of its previous height. (i) To what height does the ball rise after it hits the floor for the n-th time? How long does it take to completely stop? (ii) Find the total vertical distance the ball travels before coming to rest.

Answer 1

Answer:

i) $h = H \cdot 3^{-n}$, ii) $s = H\cdot (1 + 2\cdot \Sigma_{i = 1}^{n} 3^{-i})$

Explanation:

i) After each bounce, two thirds of previous energy is lost by the ball. Then, the height observes a geometric progression which is described herein:

$h = H\cdot \frac{1}{3^{n}}$

$h = H \cdot 3^{-n}$

Where n is the number of bounces.

ii) The total vertical distance that ball travels before coming to rest is:

$s = H + 2\cdot H \cdot \Sigma_{i = 1}^{n} \cdot 3^{-i}$

$s = H\cdot (1 + 2\cdot \Sigma_{i = 1}^{n} 3^{-i})$