Question
In $a$ one dimensional collision between two identical particles $A$ and $B, B$ is stationary and $A$ has momentum $p$ before impact. During impact, $B$ gives impulse $J$ to $A.$

Answer

Let $u=$ speed of A before impact. Thus, $p=m u$

Let $v_{1}, v_{2}=$ speeds of $A$ and $B$ after impact.

$u=v_{1}+v_{2}$ and $v_{1}-v_{2}=-e u$

$u=v_{1}+v_{2}$ and $v_{1}-v_{2}=-e u$

$\therefore v_{1}=\frac{1}{2} u(1-e)$ and $v_{2}=\frac{1}{2} u(1+e)$

$J=m v_{2}=m\left[\frac{1}{2} u(1+e)\right]=\frac{1}{2} p(1+e)$

$\Rightarrow e=\frac{2 J}{p}-1$

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