# Using the COM frame for momentum in Special Relativity

I'm having difficulty understanding 4-momentum in relativity. The problem I have is that two particles of equal mass are colliding, with one particle moving and one at rest. I know the three-momentum of the moving particle, and I need to find the total energy in the COM frame.

What's confusing me is that it appears I can completely ignore the initial frame. Using the Energy-momentum relation, I can say that the total momentum will be zero (since both particles will have opposite momentum), then I end up with:

$$E^2=m^2$$

Which I can solve because I know the mass of the two particles. This doesn't seem right since it would mean that that the energy total in the COM frame doesn't depend on the velocity of either of the particles.

I also tried a different approach where I tried to velocity of the COM frame. The velocity of the center of mass in the initial frame should be $$v/2$$, but when I use the Lorentz Velocity transformation I end up with the velocity of the rest particle being $$v/2$$, and the velocity of the moving particle being:

$$\ v'=\frac{v-v/2}{1-v*v/2}$$

Which does not simplify to $$v/2$$ as far as I can tell. Can someone please help me understand what is going on?

Regarding the velocities. They will not be equal to $$v/2$$, but they must be equal.