Threshold Energy Calculation for Proton-Proton Collision

[FreeAnnyong]

Homework Statement

When a beam of high-energy protons collides with protons at rest in the laboratory (e.g., in a container of water or liquid hydrogen, neutral pions are produced by the reaction p+p --> p+p+(pion). Compute the threshold energy of the protons in the beam for this reaction to occur.

Homework Equations

I don't even know where to start with this one.

The Attempt at a Solution

All attempts I've made are ridiculous because I didn't even know what equations to start with.

 

[kamerling]

You still have conservation of momentum and mass/energy, but you have to use the relativistic equations for them. the momentum for a particle is

$$\gamma m_0 v$$ and the kinetic energy is $$\gamma m_0 c^2$$.

It's probably easiest to work in a frame where the initial momentum is 0. In this frame the final speed of all the particles involved can be 0. You'll have to use the relativistic velocity addition formula to compute what the initial speeds in the lab frame must have been.

 

[FreeAnnyong]

I've been trying your suggestion, and I'm still getting nowhere. I try setting up the conservation of energy equation but I keep ending up with everything canceling and I just get 0=0. I feel like a complete idiot, but I can't figure out what I'm doing wrong.