A particle of mass $m$ moving with a speed $v$ collides elastically with another particle of mass $2 m$ on a horizontal circular tube of radius $R$ , then select the correct alternative(s)

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The time after which the next collision will take place is $\frac{2 π R}{v}$

The time after which the next collision will take place is proportional to m

The time after which the next collision will take place is inversely proportional to m

The time after which the next collision will take place is independent of the mass of the balls.

answer is A, D.

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Detailed Solution

Let after collision, the velocity of 2m and m be and $v_{1}$ respectively, then
$e = \frac{v_{2} - v_{1}}{v}$
$\Rightarrow v_{2} - v_{1} = e v$ …(1)
Relative velocity $v_{r} = v_{2} - v_{1}$
Time after which the next collision takes place is
$t = \frac{2 π R}{v_{r}} = \frac{2 π R}{v_{2} - v_{1}} = \frac{2 π R}{e v} \Rightarrow t = \frac{2 π R}{v}$
[ $∵$ collision is elastic $e = 1$ ]