 Solution

The initial total momentum is zero, so the outgoing neutron recoils with momentum magnitude $E_{\gamma }/c$. Put $W=(m_p+m_{\pi })c^2$. Conservation of relativistic energy givesFor example the rest energies $938.2721$, $139.5704$ and $939.5654\mathrm{MeV}$ for the proton, charged pion and neutron give this value.^[ref] The result is smaller than the initial-to-final rest-energy difference of about $138.3\mathrm{MeV}$, because some released energy becomes neutron recoil kinetic energy. Atomic binding of a captured pion is neglected in the stated stationary-particle model.