A neutron travelling with a velocity u and kinetic energy E collides elastically head on with the nucleus of an atom of mass number A at rest. The fraction of total energy retained by the neutron is:

Let mass of neutron = m (say) then mass of nucleus = mA
In head on elastic collision, velocity of neutron (after collision)
${\mathrm{v}}_1=\left(\frac{\mathrm{m}-\mathrm{mA}}{\mathrm{m}+\mathrm{mA}}\right)\mathrm{v}=\left(\frac{1-\mathrm{A}}{1+\mathrm{A}}\right)\mathrm{v}$
The fraction of total energy retained by the neutron
$=\frac{\frac{1}{2}{\mathrm{mv}}_1^2}{\frac{1}{2}{\mathrm{mv}}^2}={\left(\frac{1-\mathrm{A}}{1+\mathrm{A}}\right)}^2={\left(\frac{\mathrm{A}-1}{\mathrm{A}+1}\right)}^2$