Special relativity - kinematics

[Aleolomorfo]

Homework Statement

A photon hits a proton at rest in the laboratory frame and there is the process:
$$\gamma + p \rightarrow n+\pi^+$$
The mass of the pion is $m_\pi$ and assuming that the masses of the proton and the neutron are the same ($m$):

1. Finding the threshold energy of the foton;
2. At the threshold energy, finding the mean lifetime of the pion in the laboratory frame knowing that the mean lifetime at rest is $\tau_0$

Homework Equations

The Attempt at a Solution

For the first point I have used the invariance of the total momentum squared bewtween the lab frame before the collision and the CM frame after the collision (particle at rest) and my result is:
$$E=m_\pi + \frac{m_\pi^2}{2m}$$
I think this is correct.
For the second part I need a confirmation about my reasoning. I need to calculate the pion's velocity in the lab and then $\tau=\gamma \tau_0$. Since at the minimum energy the particles are at rest in the CM frame, is it correct to state that in the lab frame they move with the same velocity along the same direction(both to the right or equivalently to the left)? If this assumption is correct I can use the conservation of the energy $E+m=m\gamma+m_{\pi}\gamma$. From this relation I can find $\gamma$ and so $\tau$. Is it correct?

 

[TSny]

It all looks correct to me.

 

[Aleolomorfo]

It all looks correct to me.

Perfect, thank you!