First slide

Coefficient of restitution

Question

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 = $T_{1}$ and again at time t = $T_{2}$ . lf e is the coefficient of restitution, the ratio $\frac{T_{2}}{T_{1}}$ is

Difficult

A

$\frac{2}{e}$
B

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

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

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

Solution

$T_{1} = \frac{πR}{u_{1}}$ -------(1)
Time taken to collide A and B again:
$T_{2} - T_{1} = \frac{2 πR}{v_{2} - v_{1}} \Rightarrow T_{2} - T_{1} = \frac{2 πR}{{eu}_{1}} - - - - (2)$
(2) + (1) $\Rightarrow \frac{T_{2}}{T_{1}} = \frac{2 + e}{e}$

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