What is the Threshold Energy of a Proton Beam to Produce 100 GeV Pions?

[Kyrios]

Homework Statement

I am trying to find the minimum energy for a proton beam to make pions that have 100Gev of momentum. The proton beam is hitting a fixed target.

Homework Equations

p + n -> n + n + $π^{+}$

The Attempt at a Solution

In the centre of mass frame, there is zero initial momentum. But the pion has 100GeV of momentum on the other side, so I gave the neutrons the same amount in the opposite direction to cancel it out.

Then I used E=2m$c^2$ for the initial energy and got the relativistic mass. (I wasn't sure about this)
The velocity was very close to the speed of light, and I then used relativistic velocity addition to find gamma. (this is for the proton colliding with the neutron)

I had gamma ~ 9000
In the end I had an answer of $E_{proton beam}$ ~ 8.5 TeV

Is this the right sort of value to be looking at? If necessary I can write up all my working.

 

[vela]

Isn't the 100 GeV momentum of the pion measured in the lab frame? If so, that's not the momentum of the pion in the center-of-mass frame.

 

[Kyrios]

How can I find out what the momentum is in the CoM frame? Do I still need the CoM frame?