# Decay of a pion using four momentum vector

I was stuck on the following question and I can't see what the relation of this is, The question in regard was

Determine the energy E of the muon in terms of $$m0_µ$$, $$m0_π$$ and $$c$$ when the resting Pion decays into a muon with a smaller mass and a massless neutrino.

For this, it seemed like I could utilize the four-momentum vector so here is what I tried

$$P_{pion}=P_{muon}+P_{Neutrino}$$ we do not know the four-momentum vector of the neutrino so $$P_{Neutrino}=P_{pion}-P_{muon}$$

With $$P_{pion}$$=$$\begin{pmatrix} m_{pi}*c \\ 0 \\ 0 \\ 0 \end{pmatrix}$$ and $$P_{muon}$$=$$\begin{pmatrix} E_{\mu}/c \\ p_x \\ p_y \\ p_z \end{pmatrix}$$

if we would then square the last equation we would find that $$(P_{pion})^2+(P_{muon})^2+2*P_{pion}\cdot P_{muon}$$ =$$(m_{pi}*c)^2+(m_{\mu}*c)^2-2*m*E_u=0$$ since the massless particle has no momentum squared.

I have two main questions.

the first being that I do not see how $$(P_{muon})^2=(m_{\mu}*c)^2$$. The reason for this is that this is only the rest mass of the particle right, shouldnt there also be a term included for the kinetic energy that this particle receives, it seems so simple but it doesn't click I know it is true and the answer makes sense but why.

The second question in regard is, now that I can find the energy of the muon if we solve for $$E_\mu$$ but can we then also solve for the kinetic energy, since there is no potential energy in the system it is just $$E_{\mu}-m_{\mu}*c^2$$=T right?

Anyway I hope you can help me on the right track.

Since $$P_{muon}$$ is the 4-momentum, its square magnitude is the mass-squared times $$c^2$$, which can written in terms of an observer's decomposition of $$P_{muon}$$ into a temporal part, the relativistic energy $$E_{muon}/c$$ (which includes the kinetic energy) and the spatial part, the relativistic momentum $$\vec p_{muon}$$, where $$P_{muon}^2=m_{muon}^2c^2=(E_{muon}/c)^2-\vec p_{muon}\cdot\vec p_{muon}.$$
Yes, the observer determines the relativistic kinetic energy of the muon as $$T_{muon} = E_{muon} -m_{muon}c^2$$.