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The 'e' Equation

A particle of...

Question

A particle of mass $m_{1}$ moving at certain velocity collides elastically head on with a particle of mass $m_{2}$ at rest. After collision their velocities will be in the ratio of

A

\frac{m_{1} - m_{2}}{m_{1} + m_{2}}

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B

\frac{m_{1} - m_{2}}{2 (m_{1} + m_{2})}

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C

\frac{2 m_{1}}{m_{1} - m_{2}}

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D

\frac{m_{1} - m_{2}}{2 m_{1}}

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Solution

The correct option is D $\frac{m_{1} - m_{2}}{2 m_{1}}$
Using linear momentum conservation
$m_{1} v_{1} + m_{2} v_{2} = m_{1} v$ ..........(1)
$e = 1$

i.e $\frac{v_{2} - v_{1}}{0 - v} = - 1$

$v_{2} - v_{1} = v$ ...........(2)
using (1) & (2)
$m_{1} v_{1} + m_{2} v + m_{2} v_{1} = m_{1} v$
$(m_{1} + m_{2}) v_{1} = (m_{1} - m_{2}) v$

$v_{1} = \frac{(m_{1} - m_{2}) v}{(m_{1} + m_{2})}$

$v_{2} = \frac{2 m_{1} v}{(m_{1} + m_{2})}$

$v_{1} : v_{2} = \frac{m_{1} - m_{2}}{2 m_{1}}$

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