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Q.

Two equal spheres A and B lie on a smooth horizontal circular groove at opposite ends of a diameter. At time t = 0, A is projected along the groove and it first impinges on B at time t = T₁ and again at time t = T₂ . If e is the coefficient of restitution the ratio of T₂/T₁ is

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a

$\frac{2 (e + 1)}{e}$

b

$\frac{(2 + e)}{2}$

c

$\frac{2}{e}$

d

$\frac{(2 + e)}{e}$

answer is D.

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Detailed Solution

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$\begin{array}{l} T_{1} = \frac{πR}{u_{1}} \dots \dots . (1) \\ \frac{v_{2} - v_{1}}{u_{1}} = e \Rightarrow v_{2} - v_{1} = {eu}_{1}; t = T_{1} \end{array}$
Time taken to collide A and B again is $T_{2} - T_{1} = \frac{2 πR}{v_{2} - v_{1}} \Rightarrow T_{2} - T_{1} = \frac{2 πR}{{eu}_{1}}$
Dividing (ii) by (i) , we get $\frac{T_{2}}{T_{1}} = \frac{2 + e}{e}$

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