Question

Two identical 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 at time t = T, it impinges on sphere B. If e is the coefficient of restitution, the second impact will occur after a time equal to

• A. $T$
• B. $eT$
• C. $\frac{2T}{e}$
• D. $2eT$

Answer 1

The correct option is C

$\frac{2T}{e}$

If sphere A is projected with velocity v, time taken by it to strike B is equal to $\frac{\pi r}{v}=T\Rightarrow \pi r=Tv$.
Now, the coefficient of restitution is given by $e=\frac{v_B-v_A}{v}$ where $v_A$ and $v_B$ are the velocities of A and B after the collision.
Thus, $v_B-v_A=ev$.
The spheres travel with this relative velocity. It is clear that one will overtake the other after travelling a distance $2\pi r$.
$\therefore$ Time taken $=\frac{2\pi r}{v_B-v_A}=\frac{2\pi r}{ev}=\frac{2Tv}{ev}=\frac{2T}{e}$
Hence, the correct choice is (c).