% Bernd Berg Nov 01 2017.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2017 
Solutions} \medskip

\centerline{\bf Set 8} 

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\item {\bf 28. Spaceship using $(d\beta)_S$ and $(d\beta)_E$.}
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Using unit with $c=1$ we have $(d\beta)_S=g\,d\tau$ and find from
the addition theorem of velocities
$$ (d\beta)_E=\frac{\beta+(d\beta)_S}{1+\beta\,(d\beta)_S}-\beta=\beta\,
   (1-\beta\,(d\beta)_S)+(d\beta)_S-\beta=(1-\beta^2)\,(d\beta)_S\,.$$
After separation of variables and defining $\beta=\beta_E$ and $\beta'$,
$d\beta'$ the corresponding integration variables, we have
$$ \int_0^{\beta}\frac{d\beta'}{1-\beta'^2}=g\,\int_0^{\tau} d\tau'\,.$$
Integrating both sides with Mathematica we find
$$ \frac{1}{2}\,\ln\left(\frac{1+\beta}{1-\beta}\right) = g\,\tau~~
   \Rightarrow~~\beta = \tanh(g\tau)~~{\rm and}~~\gamma=\cosh(g\tau)\,.$$
The rest of the problem is solved along the previous lines:
$$  \int_0^t dt' = \int_0^{\tau} \gamma\,d\tau' = 
    \int_0^{\tau} \cosh(g\tau')\,d\tau' $$
and so on.
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