How Can Special Relativity Connect Momentum and Velocity Changes?

[ehrenfest]

Homework Statement

I am trying to solve produce an equation and the last step requires that

\frac{ \partial {m \mathbf{v}}}{\partial {s} } = \frac{ \partial { \mathbf{p}}}{c \partial {t} }

where ds is the infinitesimal Lorentz invariant length, v is velocity, and p is momentum

The problem is that I do not know how to justify this last step.

Homework Equations

The Attempt at a Solution

I tried rewriting ds in terms of tau, but then it seemed that this proper time to be equal to non-proper time which is not true.

 

[Irid]

Well, why haven't you tried to write ds in terms of dt? This gives

ds = \sqrt{(c\,dt)^2-x^2} = \sqrt{c^2-v^2}\,dt

Plug this in and you have a very easy differential equation to integrate,