Center of momentum for particle collision

[quietrain]

Homework Statement

Consider a collision in which a stationary particle with mass M is bombarded by a particle with mass m, speed v₀, and total energy (including rest energy) E_m.

PART A: Use the Lorentz transformation to write the velocities v_m and v_M of particles m and M in terms of the speed v_cm of the center of momentum.

PART B: Use the fact that the total momentum in the center-of-momentum frame is zero to obtain an expression for v_cm in terms of m, M, and v₀.

The Attempt at a Solution

ok i found the part A after like a gazillion of tries

but the problem is part B. i have totally no idea... i tried to use mv_m=mv_M since the total momentum is 0, but i don't get the ans. almost all my answers are monstrous looking expressions

help greatly appreciated!

PS: part C was adding part A and B together to get the available energy for particle collision which i got the answer from my textbook but my textbook doesn't show how to derive it. would be great if someone could tell me how to combine part A and B to show this part C thanks

 

[quietrain]

wow... i found the answer after crazy amounts of trial and errors... but i still don't understand why

my solution was, since total momentum = 0 in the center of momentum frame, then the initial momentum = 0 from conservation of momentum.

so my equation was (gamma)(m)(v₀-v_cm) - (Mv_cm) = 0

so making v_cmthe subject, i find that it is = mv₀ / {(M/gamma) + m}

but the problem is the answer's gamma is just using 1/ sqrt(1 - (v₀/c)²), but shouldn't the gamma, which is the factor in special relativity, be using v₀-v_cm instead of just v₀?