$a)\;\sqrt {\large\frac{2(M+m) \Delta E}{Mm}}$

$b)\;\sqrt {\large\frac{2(M-m) \Delta E}{Mm}}$

$c)\;\sqrt {\large\frac{2M \Delta E}{Mm}}$

$d)\;zero$

$b)\;\sqrt {\large\frac{2(M-m) \Delta E}{Mm}}$

$c)\;\sqrt {\large\frac{2M \Delta E}{Mm}}$

$d)\;zero$

Using conservation of momentum before and after collision

$ mv+M(\partial)=(M+m)v'$

$v'=\large\frac{mv}{M+m}$

now kinetic energy of the combined mass is

$\large\frac{1}{2} \bigg(\large\frac{mM}{M+m}\bigg)$$v^2=\Delta E$

$=>v^2=2 \Delta E \bigg[\large\frac{M+m}{mM}\bigg]$

$v=\sqrt {\large\frac{2 \Delta E(M+m)}{mM}}$

Hence a is the correct answer.

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