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  4. 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 T2/T1 is <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/physics/f9131b0dafdb7f667fabc39dc9299796-.png />

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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 $t = T_{1}$ and again at time $t = T_{2}$. If e is the coefficient of restitution, the ratio $T_{2}/T_{1}$ is
image

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Solution:

$T_{1}=\frac{\pi R}{u_{1}}\,\,\,(1)$
image
Time taken to collide $A$ and $B$ again:
$T_{2}-T_{1}=\frac{2 \pi R}{v_{2}-v_{1}}$
$\Rightarrow T_{2}-T_{1}=\frac{2 \pi R}{e u_{1}}\,\,\,(2)$
$(2)+(1) \Rightarrow \frac{T_{2}}{T_{1}}=\frac{2+e}{e}$