Astrophysics - Special Relativity

[Sleeker]

Homework Statement

2) If a relativistic rocket has a proper acceleration alpha that
increases with proper time tau according to:
alpha(tau) = 2/[Cosine(tau)^2 - Sine(tau)^2]
find its motion, r(t), from the point of view of a control tower
for whom the rocket is motionless at r(0) = 0.
(Hint: alpha(tau) here is the derivative with respect to tau of
ln[tan(tau + pi/4)] .)

Homework Equations

1. R=Rapidity
2. tanh(R)=β
3. d/dτ(R)=α

The Attempt at a Solution

Using formula #3 and the hint, I have R. Using formula #2 and my TI-89, I got:

(1-β)/2 = cos[t*(sqrt(1-β^2)+pi/4]^2

Using a couple of trig formulas, I have

β-1 = sin(2*t*sqrt(1-β^2))

I'm stuck there. As far as I know, there is no way to solve for β, and thus for the velocity 'v', which means I can't integrate to find r(t).

 

[vela]

Try using

$$\tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}}$$.

 

[Sleeker]

Try using

$$\tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}}$$.

I wind up at the same spot.

 

[vela]

Do it by hand, and show your work here.