Associated with each rubber ball is a bounce coefficient b. When the ball is dro

Question

Associated with each rubber ball is a bounce coefficient b. When the ball is dropped from a height h, it bounces back to a height of b h. Suppose that the ball is dropped from an initial height h, and then is allowed to bounce forever.
Use the expansion of 1/(1 - x) in powers of x to help come up with a clean formula that measures, in terms of b and h, the total up-down distance the ball travels in all of its bouncing.

Explanation / Answer

1/(1-x) = 1 + x + x^2 + x^3 + x^4 ....... distance traveled by ball,d = h+2bh+2b^2h+2b^3h+2b^4h....... =>d = h + 2h(b+b^2+b^3+b^4.....) = h + 2h[(1/(1-b)) -1] = h + 2h/(1-b) - 2h =>d=[2h/(1-b)]-h = h[2/(1-b) - 1] = h[(2-1+b)/(1-b)] = h(1+b)/(1-b) Ans.

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