\section{Relative Velocity}
The first weirdness occurs if we consider multiple people running in multiple directions and speeds. Let's explore this.

You shouldn't have to do any math here besides thinking about angles.

\problem{}
Using the given diagram, verify that if, in Bob's reference frame, Alice is running away from Bob at speed $v$, then in Alice's reference frame, Bob is running away at speed $v$.

{\em Hint: slopes}

% SPACETIME DIAGRAM - LORENTZ BOOST
\begin{center}\begin{tikzpicture}[scale=1.8]
\message{Lorentz boost^^J}

\def\xmax{2}
\def\xmaxp{2.2} % maximum of rotated axis
\def\Nlines{5} % number of world lines (at constant x/t)
\pgfmathsetmacro\ang{atan(1/2)} % angle between x and x' axes
\pgfmathsetmacro\d{0.9*\xmax/\Nlines}refer % grid size
\pgfmathsetmacro\D{\d/cos(\ang)/sqrt(1-tan(\ang)^2)} % grid size, boosted
\coordinate (O) at (0,0);
\coordinate (X) at (\xmax+0.2,0);
\coordinate (T) at (0,\xmax+0.2);
\coordinate (X') at (\ang:\xmaxp+0.2);
\coordinate (T') at (90-\ang:\xmaxp+0.2);

% WORLD LINE GRID
\message{  Making world lines...^^J}
\foreach \i [evaluate={\x=\i*\d;}] in {1,...,\Nlines}{
\message{  Running i/N=\i/\Nlines, x=\x...^^J}
\draw[world line]   (-\x,-\xmax) -- (-\x,\xmax);
\draw[world line]   ( \x,-\xmax) -- ( \x,\xmax);
\draw[world line t] (-\xmax,-\x) -- (\xmax,-\x);
\draw[world line t] (-\xmax, \x) -- (\xmax, \x);
}

% BOOSTED WORLD LINE GRID
\message{  Making world lines for boosted frame...^^J}
\fill[mydarkred,opacity=0.05]
(O) --++ (\ang:\xmaxp) --++ (90-\ang:\xmaxp) --++ (\ang:-\xmaxp) -- cycle;
\fill[mydarkred,opacity=0.05]
(O) --++ (\ang:-\xmaxp) --++ (90-\ang:-\xmaxp) --++ (\ang:\xmaxp) -- cycle;
\foreach \i [evaluate={\x=\i*\D;}] in {1,...,4}{
\message{  Running i/N=\i/\Nlines, x=\x...^^J}
\draw[world line'] (\ang:-\x) --++ (90-\ang:-\xmaxp);
\draw[world line'] (90-\ang:-\x) --++ (\ang:-\xmaxp);
\draw[world line'] (\ang:\x) --++ (90-\ang:\xmaxp);
\draw[world line'] (90-\ang:\x) --++ (\ang:\xmaxp);
}

% AXES
\draw[->,thick] (0,-\xmax) -- (T) node[left=-1] {Alice: $ct$};
\draw[->,thick] (-\xmax,0) -- (X) node[below=0] {$x$};
\draw[->,thick,mydarkred] (90-\ang:-\xmaxp) -- (T')
node[right=5,above=-1] {Bob: $ct'$};
\draw[->,thick,mydarkred] (\ang:-\xmaxp) -- (X') node[right=-1] {$x'$};

% ANGLES
\draw pic[->,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = X--O--X'};
\draw pic[<-,"$\theta$",draw=black,angle radius=34,angle eccentricity=1.2] {angle = T'--O--T};

% % PHOTON
% \draw[photon] (O) --++ (4*\d, 4*\d);
\end{tikzpicture}\end{center}
\begin{solution}
Extend the graph to the left and look at the slope of Alice's world line in Bob's reference frame. You'll find that Alice's slope in Bob's reference frame is the opposite of Bob's slope in Alice's reference frame.
\end{solution}

\problem{}
What if Alice runs to the left at speed $v$, Bob stays still, and Charlie runs to the right at speed $v$? 

From Charlie's reference frame, is Alice running away at speed $2v$? Why or why not? 
{\em Think about the consequences.}

\begin{solution}
If $v > c/2$, then this poses an immediate problem. Alice has to move at a speed below $2v$. So velocities do not just add. 
\end{solution}
\vfill
\pagebreak