Finding the velocity of an emitted particle from a decay

[HarryO]

Homework Statement

A rho meson, rest mass of 775.5 MeV at rest decays into a pion, rest mass of 139.6 MeV and a gamma ray with 0 rest mass, find the velocity of the pion produced in the decay

Relevant Equations

E=pc
E^2 = p^2*c^2 + m^2*c^4

So I know that the total energy of the system initially is 775.5MeV, because the meson is at rest. Also by conservation of energy I know that the total final energy of the system is the same thing. I also know that the initial momentum of the system is 0 because the particle is at rest. This means that the total momentum of the final state must be 0 as well, which I think means that the mometym of the gamma ray and pion must be equal and opposite. So I get.

E_ϒ = pc
p = ϒm_πv_π

ϒm_πv_π = E_ϒ/c

However from here I am stuck because I do not know how to find the energy of the gamma ray.

 

[PeroK]

Homework Statement:: A rho meson, rest mass of 775.5 MeV at rest decays into a pion, rest mass of 139.6 MeV and a gamma ray with 0 rest mass, find the velocity of the pion produced in the decay
Homework Equations:: E=pc
E^2 = p^2*c^2 + m^2*c^4

So I know that the total energy of the system initially is 775.5MeV, because the meson is at rest. Also by conservation of energy I know that the total final energy of the system is the same thing. I also know that the initial momentum of the system is 0 because the particle is at rest. This means that the total momentum of the final state must be 0 as well, which I think means that the mometym of the gamma ray and pion must be equal and opposite. So I get.

E_ϒ = pc
p = ϒm_πv_π

ϒm_πv_π = E_ϒ/c

However from here I am stuck because I do not know how to find the energy of the gamma ray.

You need to combine the equations for conservation of momentum and conservation of energy.

 

[Abhishek11235]

A systematic way is to use 4-vectors. We have:
$$P_\rho =(E,0,0,0)$$
$$P_\pi =(E',p',0,0)$$
$$P_\gamma= (E",p",0,0)$$

Now use the conservation laws to relate the quantities and don't forget to use:

$$E^2-p^2=m^2$$ or $$P_\gamma+P_\pi=P_\rho$$ for each of above relation.

Here I have taken c=1

 

[HarryO]

Okay thank you! I think I have it now.