First slide

Collisions

Question

A ball collides with a fixed inclined plane of inclination θ after falling through a distance h . If it moves horizontally just after the impact the coefficient of restitution is

Moderate

A

$\tan^{2}$ θ
B

tan θ
C

cot θ
D

$c o t^{2} θ$

Solution

Writing coefficient of restitution equation
$e = \frac{v_{2} - v_{1}}{u_{1} - u_{2}} i n i t i a l v e l o c i t y b e v_{0}; r e s o l v e i n t o c o m p o n e n t s, a s v_{0} \cos θ p e r p e n d i c u l a r t o t h e p l a n e d o w n w a r d = u_{1} a n d v_{0} \sin θ p a r a l l e l t o t h e p l a n e d o w n w a r d f i n a l v e l o c i t y b e v; r e s o l v e i n t o c o m p o n e n t s, a s v \cos θ p a r a l l e l t o t h e p l a n e a n d v \sin θ p e r p e n d i c u l a r t o t h e p l a n e u p w a r d = v_{1} s u b s t i t u t e l e g i b l y e = \frac{0 - v \sin θ}{- v_{0} \cos θ - 0} e = \frac{v}{v_{0}} \tan θ - - - (1)$

The velocity of ball will remain unchanged in the direction of sloping surface (common tangent)
$V_{0} \sin θ = v \cos θ$
$\begin{array}{l} \Rightarrow \frac{v}{v_{0}} = \tan θ - - - (2) \\ s u b s t i t u t e e q n (2) i n e q n (1) \\ \Rightarrow e = \tan^{2} θ \end{array}$

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