A neutron travelling with a velocity $v$ and kinetic energy $E$ collides head on elastically with the nucleus of an atom of mass number $A$ at rest. The fraction of its kinetic energy retained by the neutron even after the collision is

The correct option is A ${\left(\frac{A-1}{A+1}\right)}^2$
Let mass of the neutron be $m$.

Velocity of neutron before collision $u_1=v$

So, its initial kinetic energy $E=\frac{1}{2}m{v}^2$

Mass of the nucleus $M=mA$

Its velocity before collision $u_2=0$

Velocity of neutron just after collision $v_1=\frac{(m_1-e{m}_2)u_1+(1+e)m_2{u}_2}{m_1+m_2}$

where $m_1=m$, $m_2=mA$, $u_2=0$, $e=1$ and $u_1=v$

$\therefore$ $v_1=\frac{(m-mA)v+(1+e)\left(mA\right)\left(0\right)}{m+mA}=\frac{(1-A)}{(1+A)}v$

Thus kinetic energy of neutron after collision $E^{\prime }=\frac{1}{2}m{v}_1^2=\frac{1}{2}m{v}^2(\frac{1-A}{1+A}{)}^2$

So, fraction of kinetic energy retained $\frac{E^{\prime }}{E}=(\frac{1-A}{1+A}{)}^2$