How to propagate the uncertainty of momentum through the relativistic velocity equation?

Question

I have the standard deviation of momentum as $σ_p$ and I am trying to find $σ_v$. The equation I want to propogate uncertaintiy through is the relativistic velocity equation: $v=\sqrt{(p/m)^2/(1+(p/m)^2)}$. How do I do this?

Answer 1

You can just compute $\frac {dv}{dp}$. As long as the error in momentum is small, the first derivative is a good representation, so $\sigma_v \approx \frac {dv}{dp}\sigma_p$. The other usual technique is to take the log first, then take the differential. You then get results in fractional errors.