A ball is thrown down from a height of 10 m. Everytime it bounces, the hieght of the ball is half. If its height on the last bounce is 0.25 m, what is the total distance traveled by the ball.

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The ball is initially dropped from a height of 10 m. Its maximum height is half every time it bounces and on its last bounce its height is 0.25 m. The total distance traveled by the ball taking only the fact that it rises up to half the height it is dropped from is given by:

10 + 5 + 5 + 2.5 + 2.5 + 1.25 + 1.25 + 0.625 + 0.625 + 0.3125 + 0.3125 + ...

When the ball drops from 0.3125 m it should rise to 0.15625 m but the height of the ball on the last bounce is 0.25 m. An assumption can be made that after it falls from 0.625 m it only rises to 0.25 m and then stops. Making the appropriate changes in the sum for the total distance traveled by the ball gives:

10 + 5 + 5 + 2.5 + 2.5 + 1.25 + 1.25 + 0.625 + 0.625 + 0.25 + 0.25

= 29.25

The ball travels a total distance of 29.25 m

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