EPP-Problem --- Id:: QUE/EPP-01006

The mass of pion can be determined from measurements of kinetic energy of neutron in the reaction     \[ \pi^- + p  \longrightarrow n + \pi^0.\] The \(\pi^-\) meson falls in Bohr orbit, (S state), before being captured by the proton. Neglecting the kinetic energy of the pion show that \begin{eqnarray}            T_n &=& \sqrt{\omega^2 + M_n^2} - M_n\\         \omega &=& \frac{1}{2}\Big[  M_p + m_{\pi^-} -                    \frac{M_n^2}{M_p+ M_{\pi^-}} \Big]. \end{eqnarray}
The kinetic energy of the neutron is found to be \(T_n=8.872 \)MeV. Use this data to determine the mass difference \(m_{\pi^-}-m_{\pi^0}\)