# Velocity in terms of proper time

1. Mar 25, 2014

Two questions, based on the same situation: in
http://physics.stackexchange.com/questions/34204/relativistic-acceleration-equation
(question A) it is mentioned that, for an object with a constant acceleration g$_{M}$, and with $\tau$$_{0}$ =1/g$_{M}$ , after proper time $\tau$, the coordinates are
x= cosh($\tau$/$\tau$$_{0}$)
t = sinh($\tau$/$\tau$$_{0}$)
and that therefore
v = tanh($\tau$/$\tau$$_{0}$)
My first reaction was that this should be coth (position/time, cosh/sinh), but then I figured that tanh comes from v=dx/dt =d(cosh ...)/d(sinh ...) = sinh.../cosh..... Is this correct?
(question B) However, this is the end velocity after d$\tau$. So, I presume one would need to call this v= dv. If we want the end velocity after a finite amount of time, I presume integration would be in order, but since it is the rapidities that add rather than the velocities, I am not sure how this integration would look. Or perhaps there is a simpler method to find the coordinate velocity after finite proper time $\tau$ with constant acceleration g$_{M}$? (Starting at (0,0).)
I would be grateful for anyone who can untangle me from this mess.

2. Mar 25, 2014

### Bill_K

Yes

No, v = dx/dt, as you've calculated it, is the desired instantaneous coordinate velocity at time τ.

3. Mar 25, 2014