Question

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

• A. 2t
• B. 2et
• C. 2t/e
• D. $2t/e^2$

Answer 1

Answer: (C)

2t/e

conservation of momentum just before and after the collision gives:
$m{v}_0=m{v}_1+m{v}_2$......(1)
and coefficient of restitution gives:
$v_2-v_1=-e(0-v_0)\Rightarrow v_2-v_1={ev}_0$......(2)
solving (1) and (2) we get:
$v_1=v_0\left(\frac{1-e}{2}\right)and{v}_2=v_0\left(\frac{1+e}{2}\right)$
thus, second impact will occur after a time:
$\frac{2\pi R}{v_2-v_1}=\frac{2\pi R}{e{v}_0}=\frac{t}{e}$