Solve Relativistic Collision: Homework Eqs & Attempts

[mexicanrebel]

Homework Statement

A particle a traveling along the positive x-axis of frame S with speed 0.5c decays into two identical particles, a--->b+b, both of which continue to travel on the x axis. (a) Given that ma=2.5mb, find the speed of either b particle in the rest frame of particle a. (b) By making the necessary transformation on the result of part (a), find the velocities of the two b particles in the original frame S.

Homework Equations

β=pc/E, p*p=-(mc)^2, E^2=(mc^2)^2 +(pc)2

The Attempt at a Solution

I think the initial momentum in the rest frame should look like pi'=γ^2*m_a*c(0,0,0,1-(V^2/C^2))
Which can then be re-written as pi'=(0,0,0,m_a*c). Which seems to make sense considering it says that the four momentum of object a in a's rest frame has a value of 0 for the normal 3 momentum and that the fourth time value is m*c.
I think you should then find the speed of a "b" particle in the rest frame of a, I would think by using the invarience of momentum. That is that pi*pi=pf*pf, which should allow you to solve for the speed of b in the rest frame of a. But I'm not sure If I am approaching the problem right and how to proceed from what I think I am doing right. Help would be appreciated as the book seems to offer one example that feels quite different.

 

[mexicanrebel]

Still stuck; not sure if I am lacking in understanding some of the equations and properties of what's being described or if I am missing some small obvious step or fact that will allow for one to find the solution.

 

[vela]

What's the energy of one of the b particles in the rest frame of a?