Relativistic Mechanics: Answers to Exam Questions

[Beer-monster]

Hi

I'm revising for a perticle physics exam, but am having some trouble with some past paper questions involving relativity and reactions.

The one that confused me the most was

" A cosmic ray proton of energy 10^12 eV strikes a proton in the Earth’s
atmosphere. A third particle is produced. Estimate the maximum energy of a particle that might be produced from the collision, and suggest a possible particle."

I tried to work it out by equation the invariant mass to the total mass of the products since the condition has to be met for the reaction to take place.

I'd earlier worked out that the invariant mass for a particle colliding with a stationbary particle of equal mass was.

M = 2m^2 + \frac{2Em}{c^2}

Since the product mass is 2m is would stand to reason that the mass of the third particle is 2Em/c^2. But I don't get the right answer when I plug in the numbers (right outta the databook).

Another question that bugged me was that a photon was incident on a stationary proton with just the threshold energy. Describe the motion of the particles.

I know the both move off together in the direction of the incident photon but I don't remember why? Also is calcuating the energy and momentum merely a matter of uysing the standard formula (with the gamma factor)?

These are probably easy problems, but I can't find any help in my notes (I seem to be missing some). Can anyone help?

 

[Meir Achuz]

" A cosmic ray proton of energy 10^12 eV strikes a proton in the Earth’s
atmosphere. A third particle is produced. Estimate the maximum energy of a particle that might be produced from the collision, and suggest a possible particle."
The most common particle produced is a pion (m=140 MeV).
The cm energy is W^2=2M_p*E, where E=1000 GeV.
The energies are so large, you can find the maximum pion cm momentum by p_pi=(1/2)W.
Then Lorentz transform this to the lab system.
The gamma of the cm system is gamma=E/(W).