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.