# A sphere of mass m moving with constant velocity $V$ collide head on with another stationary sphere of same mass. If e is the coefficient of restitution then the ratio of the final velocities of the first and second sphere is

$\begin {array} {1 1} (a)\;\frac{1+e}{1-e} & \quad (b)\;\frac{1-e}{1+e} \\ (c)\;\frac{e}{1-e} & \quad (d)\;\frac{1+e}{e} \end {array}$

Let $V_1$ and $V_2$ be the velocities after collision in the same direction by momentum conservation
$mV_1+mV_2=mV$
=>$V_1+V_2=V$
Applying Newton's law of restitution along the common normal :
$\large\frac{relative\;velocity\;of\;separation}{relative \;velocity \;of \;approach}$$=-e$
$\large\frac{V_2-V_1}{0-V_1}=-e$
$V_2-V_1=eV$
$-V_1+V_2=eV$
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$\quad 2V_2=(1+e)V$
$\qquad V_2= \bigg(\large\frac{1+e}{2}\bigg)$V
$\qquad V_1= \bigg(\large\frac{1-e}{2}\bigg)$V
$\large\frac{V_1}{V_2}=\frac{1-e}{1+e}$
edited Jun 17, 2014