How Is Momentum Calculated for a Decomposing Particle in Relativistic Physics?

[supercali]

[SOLVED] relativistic momentum

Homework Statement

a particle with mass M and v=0 decomposes to a massless particle and another particle with the mass Q.
find the momentum of the Q particle

Homework Equations

$$E^2_{cm}=m^2c^4=(\sum{E})^2-(\sum{P})^2c^2$$

The Attempt at a Solution

i tried using that equation and also tried to do some equations with the conservation of energy and momentum but i just can't figure it out
please help if you can
thanks
and also if you can explain to me the whole deal with center of mass in relativistic momentum and energy
when does the conservation of energy apply?
what happens in the center of the mass as oppose of out of it?

 

[Dick]

Energy and momentum are always both conserved. So the total momentum after the decay is zero. So the momentum of Q is equal in magnitude to the momentum of the massless particle. Now apply conservation of energy. And apply m^2*c^4=E^2-P^2*c^2 to each particle. You should basically get two equations in two unknowns.

 

[Shooting Star]

Suppose p1 is the momentum of the massless particle to the left and p2 is the mom of the particle with rest-mass Q to the right, with speed v. Let g denote gamma(v).

p1=p2=Qgv => Also, from energy consvn,

p1c + Qgc^2 = Mc^2 => (cancel c)
Qgv + Qgc = Mc. From this, the value of v comes out in terms of M, Q and c. Not difficult if you factorize and use compodendo-dividendo. Do the algebra. We get,

v/c = (M^2-Q^2)/(M^2+Q^2). Then,

p2 = Q*g*v, all expressed in terms of M, Q and c.