Astrophysics - Special Relativity

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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.
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
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