Threshold Energy Calculation for Proton-Proton Collision

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SUMMARY

The discussion focuses on calculating the threshold energy required for high-energy protons to collide with protons at rest, resulting in the production of neutral pions through the reaction p + p → p + p + (pion). Participants emphasize the necessity of applying relativistic equations, specifically conservation of momentum and energy, to solve the problem. Key equations mentioned include the relativistic momentum formula, γm₀v, and the kinetic energy equation, γm₀c². The importance of using a reference frame where initial momentum is zero is highlighted as a critical step in the calculation process.

PREREQUISITES
  • Understanding of relativistic physics concepts, including momentum and energy conservation.
  • Familiarity with relativistic equations, specifically γm₀v and γm₀c².
  • Knowledge of the relativistic velocity addition formula.
  • Basic principles of particle physics, particularly proton interactions.
NEXT STEPS
  • Study the derivation and application of the relativistic energy-momentum relation.
  • Learn how to apply the conservation of momentum and energy in particle collision scenarios.
  • Explore the relativistic velocity addition formula in detail.
  • Investigate the production mechanisms of neutral pions in high-energy collisions.
USEFUL FOR

Students and educators in physics, particularly those focusing on particle physics and relativistic mechanics, as well as researchers involved in high-energy collision experiments.

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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.
 
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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.
 
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.
 

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