A body falls from a height 'h' on a horizontal surface and rebounds. Then it falls again and again rebounds and so on. If the restitution coefficient is $\frac{1}{3}$ , the total time taken by the body to come to rest is

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Solution

The correct option is B
Total time taken by the ball to stop bouncing
$T = t_{0} + 2 t_{1} + 2 t_{2} + 2 t_{3} + \dots$ $= \sqrt{\frac{2 h}{g}} + 2 e \sqrt{\frac{2 h}{g}} + 2 e^{2} \sqrt{\frac{2 h}{g}} + \dots \dots = \sqrt{\frac{2 h}{g}} + 2 \sqrt{\frac{2 h}{g}} (e + e^{2} + \dots \dots) = \sqrt{\frac{2 h}{g}} + 2 \sqrt{\frac{2 h}{g}} \frac{e}{(1 - e)} = \sqrt{\frac{2 h}{g}} (\frac{1 + e}{1 - e}) = \sqrt{\frac{2 h}{g}} (\frac{1 + \frac{1}{3}}{1 - \frac{1}{3}}) = 2 \sqrt{\frac{2 h}{g}}$
Hence the correct choice is (b).

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