In the lab, a relativistic proton has a momentum of 1.00 x 10^-19 kg • m/s and a rest energy of 0.150 nanojoules (nJ). What is the speed of the proton in the lab? Show steps.  c = 3.00 x 10^8 m/s, mass of proton = 1.67 x 10^-27 kg.

Expert Answers

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To derive our formula, we will start with the generic formula for relativistic momentum,


where gamma is `gamma=1/sqrt(1-(v^2/c^2))` , p is momentum, m is the rest mass of the particle, and v is the velocity. I will substitute G for p/m.

Then, I will solve for v.


I square both sides to get rid of the nasty square root in the gamma. From there, most of the simplification is straightforward.









or with the G expanded,


Don't forget the square on the v. Now, we can substitute our numbers into the formula above to calculate the relative speed of the proton.


This gives v=5.87209*10^7 m/s, or .1957c.

Particles at this velocity do not behave the same as normal particles, so the Newtonian p=mv formula does not hold here.

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