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 Math Central Quandaries & Queries
 Question from Bob, a student: A ball falls from the roof, at a height of 10m. After each impact on the ground it bounces back up to 4/5 of its previous height. How many times will the ball appear in front of a rectangular window whose bottom edge is at a height of 5m, and whose top edge at a height of 6m?

Hi Bob,

On the first drop the ball passes the window once on the way down. On the first bounce the ball rises to $\frac45 \times 10 = 8 \mbox { m }$ and hence the ball passes the window once on the way down and again on the way back up then again on the way down so that's 3. How high does it go on the second bounce? Does it still bounce higher than the top of the window? Eventually it will not bounce high enough to clear the top of the window and on those trips you will only see the ball in the window once, on the way up and the way down. Keep counting and eventually the ball will not bounce high enough to clear the bottom of the window and you won't see it at all.

Penny

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