Question

Two identical spheres $A$ and $B$ lie on a smooth horizontal circular groove at opposite ends of a diameter. $A$ is projected along the groove and at the end of time $t,$ it impinges on $B.$ If $e$ is the coefficient of restitution, the second impact will occur after a time

• A. $\frac{2t}{e}$
• B. $\frac{t}{e}$
• C. $\frac{\left(\pi t\right)}{e}$
• D. $\frac{\left(2\pi t\right)}{e}$

Answer 1

The correct option is A $\frac{2t}{e}$

$\text{Before first collision the velocity of approach is}v.\text{Separation is}\pi R$
$\text{Therefore,}v\times t=\pi R$
$t=\frac{\pi R}{v}$
$\text{After first collision:}$
$\text{Velocity of separation:}e=\frac{v^{\prime }}{v}\Rightarrow v^{\prime }=ev$
$\text{For second collision:}$
$\text{Velocity of approach is v' and distance between them is}2\pi R$
$v^{\prime }\times t^{\prime }=2\pi R{t}^{\prime }=\frac{2\pi R}{v^{\prime }}=\frac{2\pi R}{ev}=\frac{2t}{e}$