\begin{problem}
\Part{a} How fast would you have to be driving a $20\U{ft}$ long limo
to fit into $15\U{ft}$ deep garage?  \Part{b} How deep would the
garage appear to the driver of the limo?  Is the limo ever really
entirely in the garage?  Explain any apparent paradoxes.

Both of the lengths given are proper lengths.
\end{problem} % Based on undergrad memories

\begin{solution}
\Part{a}
The proper length of the limo $L_0$ needs to be contracted to the
length of the garage (equations compressed for space, read right to
left, then top to bottom).
\begin{align}
  L &= \frac{L_0}{\gamma} &
  \frac{L}{L_0} &= \frac{1}{\gamma} = \sqrt{1-\p({\frac{v}{c}})^2} \\
  \p({\frac{L}{L_0}})^2 &= 1 - \p({\frac{v}{c}})^2 &
  \p({\frac{v}{c}})^2 &= 1 - \p({\frac{L}{L_0}})^2 \\
  \frac{v}{c} &= \sqrt{1 - \p({\frac{L}{L_0}})^2} = 0.661 &
  v &= \ans{0.661 c} = \ans{198\E{6}\U{m/s}} \;.
\end{align}

\Part{b}
The garage $L_{g0}$ is length contracted in the limo frame
\begin{align}
  L_g &= \frac{L_{g0}}{\gamma} = L_{g0}\sqrt{1-\p({\frac{v}{c}})^2}
       = 15\U{ft}\sqrt{1-.661^2} = \ans{11.2\U{ft}} = \ans{3.43\U{m}} \;.
\end{align}

Wait, how can the limo fit into a garage that appears even shorter
than its proper length of $15\U{ft}$?  This is a
relative-simultanaeity effect like the muon clock running slower than
an earth clock while the earth clock runs slower than the muon clock.
In the garage frame, the limo-nose-passes-crashes-into-back-wall event
$A$ and the limo-tail-passes-door event $B$ occur at the same time.
In the limo frame, event $A$ happens some time before event $B$.

Drawing a space-time diagram in the garage frame may help clarify the
different events.  The limo is the grey smear.  The red dotted lines
represent a $1\U{ls}$ time and space grid for the limo driver.  The
blue lines show the speed of light.  The green lines show the garage.
I've rescaled the garage and limo to make them $1\U{ls}$ and
$1.33\U{ls}$ long respectively so that the axes have simple labels.
The limo driver thinks that at the same time as event $A$, the tail of
the limo is back at event $C$, while the door of the garage is up at
event $D$.  Note that the garage is less than $1\U{ls}$ deep in the
limo frame.
\begin{center}
\includegraphics[height=2in]{problem.relativistic-limo.limo}
\end{center}
``Entirely in the garage'' is something of a trick question, since it
means ``all of the limo is in the garage at same time'' and ``at the
same time'' depends on your reference frame.  In the garage frame, the
limo is entirely in the garage for a single instant.  In the limo
frame, the limo is never entirely in the garage.  An absolute,
true-no-matter-what answer to the question is not possible in the
relativistic world-view.
\end{solution}
