How far does the bouncing ball travel?

Question

A ball is dropped from a height of 10 metres. Each time it hits the ground it rebounds to half the height it fell from, and this continues indefinitely. 1. What is the total distance the ball travels? 2. Does it bounce forever, and does it take forever to stop?

Accepted Answer

30 metres. It bounces infinitely many times but comes to rest in finite time. The distance. The first fall is 10 m. After the first bounce the ball rises 5 m and falls 5 m (10 m of travel), then rises 2.5 and falls 2.5 (5 m), and so on. The up-and-down pairs total 10 + 5 + 2.5 + ... = 10 / (1 minus 1/2) = 20 m. Adding the initial fall: 30 metres. A shortcut. Total distance = first drop plus twice the sum of all rebound heights = 10 + 2 × (5 + 2.5 + 1.25 + ...) = 10 + 2 × 10 = 30. For a rebound ratio r the total is h × (1 + r)/(1 minus r): with r = 1/2 that is 3h. The time. Falling from height…

How far does the bouncing ball travel?

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