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A bouncing ball

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A ball is dropped from rest at a height of $12$ m. After each bounce it rises to $\frac{2}{3}$ of the height it last fell from, and this goes on forever. What total vertical distance, in metres, does the ball travel? Give the answer as an integer.

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Hint

Apart from the first drop, every height is travelled twice, up and down.

Solution

The first drop is $12$ m.
The rebound heights are $8$ , $8 \times \frac{2}{3}$ and so on, a geometric series with sum $\frac{8}{1 - \frac{2}{3}} = 24$ .
Each rebound height is covered going up and coming down, giving $2 \times 24 = 48$ .
The total distance is $12 + 48 = 60$ .

Answer: $60$

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