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The 'e' Equation

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Question

A ball is projected form a point A on a smooth inclined plane which makes an angle $α$ to the horizontal. The velocity of projection makes an angle $θ$ with the plane upwards. If on the second bounce the ball is moving perpendicular to the plane, find e in terms of $α$ and $θ$ . Here e is the coefficient of restitution between the ball and the plane

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Solution

$F i r s t b o u n c e w i l l b e a t t h e t i m e T_{1} = \frac{2 u s i n θ}{g c o s α} S e c o n d b o u n c e w i l l b e a t t h e t i m e a f t e r t h e f i r s t b o u n c e T_{2} = \frac{2 e u s i n θ}{g c o s α} s i n c e v e l o c i t y a l o n g v d i r e c t i o n a f t e r t h e 1 s t b o u n c e i s e u s i n θ . I n t i m e T_{1} + T_{2} t h e v e l o c i t y a l o n g x d i r e c t i o n b e c o m e s z e r o t h u s, 0 = u c o s θ - g s i n α (T_{1} + T_{2}) \Rightarrow 0 = u c o s θ - g s i n α \frac{2 u s i n θ}{g c o s α} (1 + e) \Rightarrow e = \frac{c o t θ c o t α}{2} - 1$

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