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

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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$

$\large\frac{V_2-V_1}{0-V_1}=-e$

$V_2-V_1=eV$

$-V_1+V_2=eV$

________________

$\quad 2V_2=(1+e)V$

$\qquad V_2= \bigg(\large\frac{1+e}{2}\bigg)$V

$\qquad V_1= \bigg(\large\frac{1-e}{2}\bigg)$V

$\large\frac{V_1}{V_2}=\frac{1-e}{1+e}$

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