Maximum rest mass in particle collision

[Sleestak]

Homework Statement

Suppose that a certain accelerator can give protons a kinetic energy of 200 GeV. The rest mass of a proton is 0.938 Gev/c^2. Calculate the largest possible rest mass M0 of a particle that could be produced by the impact of one of the high-energy protons on a stationary proton in the following process: p+p --> p+p+x

Homework Equations

Ek = γmc^2
p=γmv
E^2 - (pc)^2=(mc^2)^2
Energy before = energy after
P before = P after

The Attempt at a Solution

Ok, I am almost positive this has to do with conservation of energy and momentum, and that is how I would end up finding the rest mass of the particle, I am just unsure of how I would know when this is at a maximum. My initial thought is that it would be when both the protons after the collision are stationary, but I have no way of proving that.

Energy before = 200 Gev +0.938 + 0.938 = 201.876 GeV
Energy before = γmc^2 = 0.938 GeV / √(1-v^2/c^2)
I solved for v and got 0.999989c, which I can use for momentum, so
P before = (0.938 GeV * 0.999c)/ √(1-(0.999989)^2) = 199.998 GeV/c

After that, I could just find equations for the energy and momentum after, but I want to know first when it would be at a maximum. My professor said it had something to do with the center of mass reference frame, but I'm not sure how that would come into play. Help?

 

[vela]

Homework Statement

Suppose that a certain accelerator can give protons a kinetic energy of 200 GeV. The rest mass of a proton is 0.938 Gev/c^2. Calculate the largest possible rest mass M0 of a particle that could be produced by the impact of one of the high-energy protons on a stationary proton in the following process: p+p --> p+p+x

Homework Equations

Ek = γmc^2
p=γmv
E^2 - (pc)^2=(mc^2)^2
Energy before = energy after
P before = P after

The Attempt at a Solution

Ok, I am almost positive this has to do with conservation of energy and momentum, and that is how I would end up finding the rest mass of the particle, I am just unsure of how I would know when this is at a maximum. My initial thought is that it would be when both the protons after the collision are stationary, but I have no way of proving that.

Stationary in what frame? As your professor suggested, analyze the interaction in the center-of-mass frame.

Energy before = 200 Gev +0.938 + 0.938 = 201.876 GeV
Energy before = γmc^2 = 0.938 GeV / √(1-v^2/c^2)
I solved for v and got 0.999989c, which I can use for momentum, so
P before = (0.938 GeV * 0.999c)/ √(1-(0.999989)^2) = 199.998 GeV/c

After that, I could just find equations for the energy and momentum after, but I want to know first when it would be at a maximum. My professor said it had something to do with the center of mass reference frame, but I'm not sure how that would come into play. Help?

 

[Orodruin]

What would happen with energy and momentum if both protons were stationary after the collision?

And just to check before making life unnecessarily complicated: Are you familiar with 4-vector notation?