A positively charged sigma particle (symbol \Sigma^{+}) produced in a particle physics experiment decays very quickly into a neutron and positively charged pion before either its energy or momentum can be measured. The neutron and pion are observed to move in the same direction as the \Sigma^{+} was originally moving, with momenta of 4702 MeV/c and 169 MeV/c, respectively. What was the kinetic energy of the \Sigma^{+} and its mass?
Strategy The decay reaction is
\Sigma^{+} \rightarrow n+\pi^{+}where n is a neutron. Obviously the \Sigma^{+} has more mass than the sum of the masses of n and \pi^{+}, or the decay would not occur. We have to conserve both momentum and energy for this reaction. We use Equation (2.70) to find the total energy of the neutron and positively charged pion, but in order to determine the rest energy of \Sigma^{+}, we need to know the momentum. We can determine the \Sigma^{+} momentum from the conservation of momentum.
E^{2}=p^{2} c^{2}+E_{0}^{2} (2.70)