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

^{1}⁄_{2}*c*_{d}*ρA*_{p}*v*^{2}and the lift-related force
^{1}⁄_{2}(*mg*)^{2}/(*ρv*^{2}*A*_{s}). 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 ^{1}⁄_{2}*c*_{d}*ρA*_{p}*v*^{2}

and
^{1}⁄_{2}(*mg*)^{2}/(*ρv*^{2}*A*_{s})
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