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Types of Collision

An object of ...

Question

An object of mass $m_{1}$ collides elastically with another object of mass $m_{2}$ , which is at rest. After the collision, the objects move with equal speeds in opposite directions. The ratio of the masses, $m_{2} : m_{1}$ is -

2 : 1

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1 : 1

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1 : 2

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3 : 1

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Solution

The correct option is D $3 : 1$
From conservation of linear momentum;

$p_{i} = p_{f}$

$\Rightarrow m_{1} u_{1} + m_{2} (0) = m_{1} (- v) + m_{2} v$

$\Rightarrow m_{1} u_{1} = v (m_{2} - m_{1}) . . . . . (i)$

Also, co-efficient of restitution,
$e = \frac{v_{separation}}{v_{approach}} = \frac{v - (- v)}{u_{1}} = \frac{2 v}{u_{1}} = 1 [For elastic collision]$

$\Rightarrow u_{1} = 2 v . . . . . (i i)$

From $(i)$ and $(i i)$ ,

$m_{1} (2 v) = v (m_{2} - m_{1})$

$\Rightarrow 2 m_{1} = m_{2} - m_{1}$

$\Rightarrow 3 m_{1} = m_{2}$

$\Rightarrow \frac{m_{2}}{m_{1}} = \frac{3}{1} = 3 : 1$

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