# Relativistic collision problem

1. Nov 21, 2008

### insynC

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

A collision between two protons can result in the creation of a positive pion and the conversion of one proton to a neutron:

p$$^{}+$$ + p$$^{}+$$ --> p$$^{}+$$ + n + $$\pi$$ $$^{}+$$

(The last one is a positive pion, again sorry about my bad use of latex.)

Calculate the minimum kinetic energy (in MeV) for the protons in this reaction if the two protons have equal energy.

2. Relevant equations

I think conservation of energy and momentum are the key to solving this question.

3. The attempt at a solution

The fact the two initial protons means that as they have the same rest mass, they will have the same momentum and so the momentum of the initial system, and hence the final system, must be zero.

Thus while maintaining the total momentum as zero, I know I have to adjust the velocities of the three final particles to minimise the total energy of the system.

As momentum is proportional to v $$\gamma$$ and energy is proportional to $$\gamma$$, my thought is that the gamma factor for the more massive particles (neutron and proton) need to be minimised whilst the gamma for the pion needs to be maximised, as conceptually this should provide the minimum energy whilst still conserving momentum.

Nonetheless actually putting this into action has not led me to any success. I'm not sure if this is the right way to approach the problem, but I have the proton and neutron heading off perpendicularly (say in an x-y plane the proton in the -x direction and the neutron in the -y direction) whilst the pion is at some angle in the first quadrant (where x & y are positive).

Trying to solve the equations though are not only horrendous, but I end up with two variables in the one equation: $$\theta$$ (angle pion makes with x axis) and the gamma factor for the pion.

Is there a better way to approach this problem?

2. Nov 21, 2008

### Dick

Since the total initial momentum is zero, at the minimum energy, won't all of the reaction products be produced at rest in the center of mass frame?

3. Nov 21, 2008

### insynC

Ah, woops. That would seem to make sense, I think I overcomplicated it :S.