Stationary particle decay into two particles with DIFFERENT masses

[mmh37]

I have been thinking and thinking this over, but I just can't find the solution - can anyone help?
A particle of mass 7m which is initially at rest in the laboratory frame decays into two fragments whose rest masses are 2m and 3m. Find the energies of the fragments and their speeds in the lab frame.
Help's much appreciated!

 

[abszero]

Well, energy will be conserved, as will momentum. So you can calculate their total energy by using E = mc^2, and then you can use conservation of momentum as well, and then you have two equations with two unknowns. Solve.

 

[mmh37]

That's what I've been trying to do:

conservation of energy: 7mc^2 = gamma(1)*2mc^2 + gamma(2)*3m^2

(different gamma factors as the particles move with different velocities)

conservation of momentum:

0 = gamma(1)*2m*v1 + gamma(2)*3m*v2


Then you end up with a very unpleasant equation, which I cannot solve.

Does anyone know how to do so or whether there is an easier way (which I'm sure has to exist). I've also tried to work with the energy momentum invariant and different frames of reference (which doesn't make any sense at all here, but I wanted to give it a go anyway).

 

[fargoth]

well, as far as i can remember the quantity $$\sqrt{m^2c^4+p^2c^2}$$ is conserved.
so $$7mc^2=\sqrt{9m^2c^4+4m^2c^4+p_1^2c^2+p_2^2c^2}$$
and the momentum is conserved too, so $$p_1=-p_2$$
and i think you can find the different velocities by $$\gamma_1 3mv=p_1$$
and $$\gamma_2 2mv=p_2$$

 

[mmh37]

thanks for your help!
It now works!