Astrophysics - Special Relativity

AI Thread Summary
The discussion revolves around solving a homework problem related to the motion of a relativistic rocket with a proper acceleration that varies with proper time. The acceleration formula provided leads to the calculation of rapidity, with participants attempting to derive the velocity and position functions. The main challenge is solving for the velocity variable β, which is crucial for integrating to find the rocket's motion r(t). Suggestions include using hyperbolic tangent identities to facilitate the calculations. The conversation highlights the complexity of the problem and the need for detailed step-by-step solutions.
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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).
 
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Try using

\tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}}.
 
vela said:
Try using

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

I wind up at the same spot.
 
Do it by hand, and show your work here.
 
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