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\centerline{\bf Special and General Relativity (PHZ 4601/560 
Fall 2018) Midterm October 17.} % \bigskip \vskip -20pt

\begin{figure}[tb]
\centerline{\hbox{ \psfig{figure=figures/1mink,width=10cm} }}
Figure: Minkowski space in which observer~A is at rest and
flashes a light signal at observer~B, who moves with speed $4/5$,
and flashes the signal back (here we use units of seconds $[s]$ 
and $c=1$ for the speed of light).
\end{figure}

\begin{enumerate}

\item {\bf Light signals and travel in two inertial frames: (50\%).}

In the above figure Minkowski space is 
parametrized by the coordinates of the rest frame $S$ of an observer~A. 
While observer A stays at rest, observer~B moves with speed $\beta=4/5$ 
along the positive $x$ axis. At their common origin both, A and B, have 
set their clocks to zero. After 15~$[s]$ observer~A emits at position 
A$_1$, i.e. at $(15\,[s],0)$ a light signal which reaches observer~B 
at position B$_0$, who flashes it back at observer~A, who receives it 
at position A$_2$ as drawn in the figure. 

\begin{enumerate}

\item Find the coordinates of $A_1$ in the rest frame $S'$ of B,
      where the translational freedom is used so that the origins 
      of the restframes of A and B agree at time $t=t'=0$.

\item Find the coordinates of $B_0$ in the rest frames
      $S$ of A and $S'$ of B.

\item Find the coordinates of $A_2$ in the rest frames
      $S$ of A and $S'$ of B.

\item Transform the above figure into the rest frame $S'$ of B.

\end{enumerate} \bigskip

{\bf Turn over to backside.} \vfil\eject

\item{\bf Spacetrip (50\%).}

\begin{enumerate}

\item Assume that the earth is in an inertial frame. A spaceship leaves 
the earth at time~0. The spaceship is constructed so that it has an 
acceleration $g$ in each of its own instantaneous rest frames. By its 
own clock, it accelerates on a straight-line path for 1~year, decelerates 
at the same rate for 1~year, turns around, accelerates for 1~year,
decelerates for 1~year, and lands on earth. Calculate the time on 
earth at the landing. 

Instructions: Use $g=9.81~[m/s^2]$, one year = $365\times 24\times 
3600~[s]$ and for the speed of light $c=300,000~[km/s]$. State the 
result in units of years and decimal fractions of years to at least 
three significant digits. 

\item How far away from Earth did the spaceship travel? Express the 
result in light years.

\end{enumerate} \bigskip

% \begin{figure}[th] % From EMscript/figures_plt/minkowski_space_ex.
% \vspace{5pc}       % 3mink-prob.plt
% \centerline{\hbox{ \psfig{figure=e3mink,width=10cm} }}
% \caption{Minkowski \label{fig_exmink3} } \end{figure}

\end{enumerate}

\end{document}
