Relativistic electrons and positrons

[mangojuice14]

Homework Statement

The question states that an electron and positron, each with rest mass energy of 511keV collide head on and create a proton and antiproton each with rest mass energy 938MeV. The question asks us to find the minimum kinetic energy of the electron and positron.

Homework Equations

Relativistic energy: E = T + mc² where T is the kinetic energy.

The Attempt at a Solution

I have solved the problem with the relativistic energy equations by setting the proton's and antiproton's final state at rest
T(electron) +m_ec² = m_pc²
and obtained the solution that the electron and positron each would need a kinetic energy of 937.5MeV.

I checked the answer and it's correct but the energy implies that the electron would be moving at 60 times light speed. Does this mean the problem is purely theoretically and is not physically possible or am I missing something. Thanks

 

[TSny]

Welcome to PF!

I checked the answer and it's correct but the energy implies that the electron would be moving at 60 times light speed.

Can you show your calculation for this? Remember, the formula for kinetic energy, T, in relativity is not (1/2)mv².
You should be able to show that an electron with any finite amount of energy (no matter how large) would still move at less than the speed of light.

 

[mangojuice14]

Can you show your calculation for this? Remember, the formula for kinetic energy, T, in relativity is not (1/2)mv².
You should be able to show that an electron with any finite amount of energy (no matter how large) would still move at less than the speed of light.

Oh right! I was actually using the small velocity approximation...woops. Using $$E = \frac {mc^2} {\sqrt{1-\frac{v^2}{c^2}}}$$ where E would be 938MeV, I now obtain a speed of 0.9999998c. Thanks!