Find relativistic momentum of electron given kinetic energy.

[oddjobmj]

Homework Statement

At what energy does an electron become “relativistic”? Consider electrons with
kinetic energies of 50 eV, 50 keV, and 50 MeV. For each case, calculate the momentum
of the electron first using the non-relativistic formula for kinetic energy, and then using
the correct relativistic formulas. Express the momentum in units of eV/c, or keV/c, or
MeV/c (whichever is appropriate), as discussed in section 2.13 of Thornton and Rex.
(For this you need to know that the rest energy of an electron is 0.511 MeV.) Compare
your answers for each case. When is it important to use the relativistic formulas?

Homework Equations

Non-relativistic:
K_e=\frac{1}{2}mv²

p=mv

Relativistic:
p=\frac{mv}{\sqrt{1-(\frac{v}{c})^2}}

The Attempt at a Solution

I was able to use the non-relativistic equations to find momentums by equating the equation for kinetic energy and momentum with the final result of:

p=\sqrt{2K_em}

When it comes to the relativistic momentum, however, I can't seem to remember how to find v! From what I remember it is straight forward but I can't find what I need. Any suggestions are welcome, thank you!

 

[Curious3141]

Have the covered the *very* useful equation $E^2 = m^2c^4 + p^2c^2$ with you? I would definitely use that if I could.

 

[oddjobmj]

It definitely looks familiar but we have since moved on to new material and I don't recall the significance / relevance of that relationship.

I take it I could just solve for p and replace E with the sum of the rest mass and the given kinetic energy?

 

[Curious3141]

It definitely looks familiar but we have since moved on to new material and I don't recall the significance / relevance of that relationship.

I take it I could just solve for p and replace E with the sum of the rest mass and the given kinetic energy?

Yes, E is the sum of the rest mass-energy and the kinetic energy. You're allowed to state the answer in eV/c, so you don't even have to do any conversions.

 

[oddjobmj]

Thank you!

So p=\sqrt{\frac{(mc^2+K_e)^2-m^2c^4}{c^2}}

I got about 7.1 MeV/c for the 50 MeV non-relativistic and ~51 MeV for the same electron using the relativistic equation above. Does that sound about right?

 

[Curious3141]

Thank you!

So p=\sqrt{\frac{(mc^2+K_e)^2-m^2c^4}{c^2}}

I got about 7.1 MeV/c for the 50 MeV non-relativistic and ~51 MeV for the same electron using the relativistic equation above. Does that sound about right?

I'm getting 50.5MeV/c for the relativistic value for the 50MeV electron.

Didn't check the non relativistic value but it should be quite badly off at that energy level.

 

[oddjobmj]

Perfect, thank you! Just wanted to make sure I understood what you were explaining.