# Decay of a particle of mass M into two particles

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1. Feb 12, 2017

### Forco

1. The problem statement, all variables and given/known data
A particle of mass M and 4-moment P decays into two particles of masses m1 and m2
1) Find the total energy of each particle (lab frame).
2) Show that the kinetic energy T1 of the first particle in the same reference frame is given by
$$T_1= \Delta M (1 - \frac{m_1}{M} - \frac{\Delta M}{2M})$$
3) A pion (M= 139.6 MeV) decays into a muon (m=105,7 MeV) and a neutrino (m=0). Find the kinetic energy of the muon and the neutrino (pion rest frame and lab frame).

2. Relevant equations
$$E_1+E_2 = Mc^2$$
$$E_{1}^{2} +\frac{p^2}{c^2} = m_{1}^{2} c^4$$
$$E_{2}^{2} +\frac{p^2}{c^2} = m_{2}^{2} c^4$$
3. The attempt at a solution
I was able to do the first one, since it was really simple, I only needed to set

$$E_{1}^{2} - m_{1}^{2} c^4 = E_{2}^{2} - m_{2}^{2} c^4$$
And since $$E_1+E_2 = Mc^2$$, it was easy to find that
$$E_1 = \frac{M^2+m_{1}^{2}-m_{2}^{2}}{2M}$$
Which is the correct expression for the energy. I'm having some trouble with the other two though. Especially the second one.
Any help is appreciated.

2. Feb 12, 2017

### vela

Staff Emeritus

Are you sure? I don't see how you can justify equations you started with in the lab frame.