# Center of momentum for particle collision

1. Apr 15, 2010

### quietrain

1. The problem statement, all variables and given/known data
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) Em.

PART A: Use the Lorentz transformation to write the velocities vm and vM of particles m and M in terms of the speed vcm 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 vcm in terms of m, M, and v0.

3. 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 mvm=mvM 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

2. Apr 18, 2010

### 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-vcm) - (Mvcm) = 0

so making vcmthe 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-vcm instead of just v0???

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