\amarginfig{t}{% produced by gnuplot < gnu.flight
\small \begin{center}
\sf thrust (kN) \hfill \,\hspace*{1cm} \\[1.5mm]
\mono%%% \raisebox{-2mm}{$v$}
{\epsfxsize=53mm\epsfbox{figs/flightv.eps}}%
{\epsfxsize=53mm\epsfbox{figs/flightvC.eps}}%
\\
\hfill speed (m/s)
\end{center}
\caption[a]{The force required to keep a plane moving,
as a function of its speed $v$,
is the sum of an ordinary
drag force $ \frac{1}{2} c_{\rm d} \rho \Ap v^2$ -- which increases
with speed --
and the lift-related force (also known as the induced drag)
$\frac{1}{2} \frac{(mg)^2}{\rho v^2 \As}$ -- which decreases
with speed.
There is an ideal speed, $v_{\rm optimal}$, at which the
force required is minimized. The force is an energy per distance,
so minimizing the force also minimizes the fuel per distance.
To optimize the
fuel efficiency, fly at $v_{\rm optimal}$.
This graph shows our cartoon's estimate of the
thrust required, in kilonewtons, for a
Boeing \ind{747}\nlabel{p747da} of mass 319\,t, wingspan 64.4\,m, drag coefficient 0.03,
and frontal area 180\,m$^2$, travelling
in air of density $\rho = 0.41\,$kg/m$^3$
% .4135
(the density at a height of 10\,km),
as a function of its speed $v$ in m/s.
Our model has an optimal speed $v_{\rm optimal} = 220\,\m/\s$ (540\,mph).
%% sausages
For a cartoon based on \ind{sausage}s, this is a good match to real life!}
\label{fig.forcesum}
}%