A small ball is projected from point A on the floor towards a vertical wall as shown in the figure. It hits the wall when its velocity is horizontal. Ball reaches point A after one bounce on the floor. If the coefficient of restitution is the same for the collisions, find its value

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0.25

0.5

0.75

0.33

answer is A.

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Detailed Solution

$T = t_{1} + t_{2} + t_{3}$

$t_{1} = \frac{u \sin θ}{g}; t_{2} = \frac{eu \sin θ}{g}; t_{3} = \frac{2 e^{2} u \sin θ}{g}$

$t = \frac{2u \sin θ}{g}$

$T = t_{1} + t_{2} + t_{3}$

$\frac{2u \sin θ}{g} = \frac{u \sin θ}{g} + \frac{eu \sin θ}{g} + \frac{{2e}^{2} u \sin θ}{g}$

$2 = 1 + e + 2 e^{2}$

$2 e^{2} + e - 1 = 0$

$2 e (e + 1) - 1 (e + 1) = 0$

$(2 e - 1) (e + 1) = 0$

$2 e - 1 = 0$

$e = \frac{1}{2} = 0.5$

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