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**[SOLVED] Relativistic kinetic energy and proton collisions**

## Homework Statement

Find the minimum proton kinetic energy required to produce an

antiproton in the reaction

[tex]P+P\rightarrow P+P+P+\overline{P}[/tex]

for protons:

(a) Incident on protons of equal and opposite momentum,

(b) Incident on stationary isolated protons.

## Homework Equations

[tex]K=(\gamma-1)mc^{2}[/tex]

[tex]E=\gamma mc^{2}[/tex]

## The Attempt at a Solution

For part (a) equal and opposite momentum would give,

[tex]2(\gamma mc^{2}=4mc^{2}[/tex]

[tex]\gamma=2[/tex]

[tex]v^{2}=(3/4)c^{2}[/tex]

So,

[tex]K=(\gamma-1)mc^{2}[/tex]

[tex]K=1.5*10^{-10} J[/tex]

For part (b) assuming only one proton has all the kinetic energy required to create a proton anti-proton pair would give,

[tex](\gamma mc^{2}=4mc^{2}[/tex]

[tex]\gamma=4[/tex]

[tex]v^{2}=(15/16)c^{2}[/tex]

So,

[tex]K=(\gamma-1)mc^{2}[/tex]

[tex]K=4.5*10^{-10} J[/tex]

Please can anyone tell me if I hae gone wrong somewhere, I'm new to all this relativistic stuff.