if the plane turned its fuel’s power into drag power and lift power perfectly
efficiently. (Incidentally, another name for “energy per distance travelled”
is “force,” and we can recognize the two terms above as the drag force
12cdρApv2and the lift-related force 12(mg)2/(ρv2As). The sum is the
force, or “thrust,” that specifies exactly how hard the engines have to push.)

Real jet engines have an efficiency of about ε = 1/3, so the energy-per-
distance of a plane travelling at speed v is (C.14)

This energy-per-distance is fairly complicated; but it simplifies greatly if
we assume that the plane is designed to fly at the speed that minimizes the
energy-per-distance. The energy-per-distance, you see, has got a sweet-
spot as a function of v (figure C.5). The sum of the two quantities 12cdρApv2
and 12(mg)2/(ρv2As) is smallest when the two quantities are equal. This
phenomenon is delightfully common in physics and engineering: two things
that don’t obviously have to be equal are actually equal, or equal within a
factor of 2.

So, this equality principle tells us that the optimum speed for the plane
is such that (C.15)

i.e. (C.16)

This defines the optimum speed if our cartoon of flight is accurate; the
cartoon breaks down if the engine efficiency ε depends significantly on
speed, or if the speed of the plane exceeds the speed of sound (330 m/s);
above the speed of sound, we would need a different model of drag and
lift.

Let’s check our model by seeing what it predicts is the optimum speed
for a 747 and for an albatross. We must take care to use the correct air-
density: if we want to estimate the optimum cruising speed for a 747 at
30 000 feet, we must remember that air density drops with increasing al-
titude z as exp(−mgz/kT), where m is the mass of nitrogen or oxygen
molecules, and kT is the thermal energy (Boltzmann’s constant times ab-
solute temperature). The density is about 3 times smaller at that altitude.

The predicted optimal speeds (table C.6) are more accurate than we
have a right to expect! The 747’s optimal speed is predicted to be 540mph,
and the albatross’s, 32mph – both very close to the true cruising speeds of
the two birds (560mph and 30–55mph respectively).

Let’s explore a few more predictions of our cartoon. We can check
whether the force (C.13) is compatible with the known thrust of the 747.
Remembering that at the optimal speed, the two forces are equal, we just Figure C.5. The force required to keep a plane moving, as a function of its speed v, is the sum of an ordinary drag force 12cdρApv2 – which increases with speed – and the lift-related force (also known as the induced drag)
 1 (mg)2 2 ρv2As
– which decreases with speed. There is an ideal speed, voptimal, 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 voptimal. This graph shows our cartoon’s estimate of the thrust required, in kilonewtons, for a Boeing 747 of mass 319 t, wingspan 64.4 m, drag coefficient 0.03, and frontal area 180 m2, travelling in air of density ρ = 0.41 kg/m3 (the density at a height of 10 km), as a function of its speed v in m/s. Our model has an optimal speed voptimal = 220 m/s (540 mph). For a cartoon based on sausages, this is a good match to real life! 