Question

A particle of mass $m$ moving with a speed $v$ hits elastically another particle of mass $2m$ at rest, on a horizontal smooth circular tube of radius $R$.

• A. Speed of $2m$ just after collision will be $\frac{2v}{3}$
• B. Speed of $m$ just after collision will be $\frac{v}{3}$
• C. Velocity of separation just after first collision will be $v$
• D. The time after which the next collision will take place is equal to $\frac{2\pi R}{v}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Speed of $2m$ just after collision will be $\frac{2v}{3}$
B Speed of $m$ just after collision will be $\frac{v}{3}$
C Velocity of separation just after first collision will be $v$
D The time after which the next collision will take place is equal to $\frac{2\pi R}{v}$

Let the final velocities of $2m$ and $m$ be $V_2$ and $V_1$ respectively.
Using conservation of momentum
$2m{V}_2+m{V}_1=mv$
$2V_2+V_1=v\dots \dots \left(1\right)$
As collision is elastic, from coefficient of restitution
$e=1=\frac{V_2-V_1}{v}$
$V_2-V_1=v\dots \dots \left(2\right)$

Adding equations $\left(1\right),\left(2\right)$
$V_2=\frac{2v}{3},V_1=\frac{-v}{3},$
When they meet again displacement of one particle relative to another is $2\pi R$
As $V_{\text{sep}}=V_2-V_1=v$
and $t=\frac{2\pi R}{V_{\text{rel}}}=\frac{2\pi R}{v}$