The range of the bird is the intrinsic range of the fuel, 4000 km, times a
factor εffuel/ (cdfA)1/2. If our bird has engine efficiency ε = 1/3 and drag-to-
lift ratio (cdfA)1/2 1/20, and if nearly half of the bird is fuel (a fully-laden
747 is 46% fuel), we find that all birds and planes, of whatever size, have
the same range: about three times the fuel’s distance – roughly 13 000 km.

This figure is again close to the true answer: the nonstop flight record
for a 747 (set on March 23–24, 1989) was a distance of 16 560 km.

And the claim that the range is independent of bird size is supported
by the observation that birds of all sizes, from great geese down to dainty
swallows and arctic tern migrate intercontinental distances. The longest
recorded non-stop flight by a bird was a distance of 11 000 km, by a bar-
tailed godwit.

How far did Steve Fossett go in the specially-designed Scaled Composites
Model 311 Virgin Atlantic GlobalFlyer? 41 467 km. [33ptcg] An
unusual plane: 83% of its take-off weight was fuel; the flight made careful
use of the jet-stream to boost its distance. Fragile, the plane had several
failures along the way.

One interesting point brought out by this cartoon: if we ask “what’s
the optimum air-density to fly in?”, we find that the thrust required (C.20)
at the optimum speed is independent of the density. So our cartoon plane
would be equally happy to fly at any height; there isn’t an optimum density;
the plane could achieve the same miles-per-gallon in any density; but
the optimum speed does depend on the density (v2 ~ 1/ρ, equation (C.16)).
So all else being equal, our cartoon plane would have the shortest journey
time if it flew in the lowest-density air possible. Now real engines’ efficien-
cies aren’t independent of speed and air density. As a plane gets lighter by
burning fuel, our cartoon says its optimal speed at a given density would
reduce (v2 ~ mg/(ρ(cdApAs)1/2)). So a plane travelling in air of constant
density should slow down a little as it gets lighter. But a plane can both
keep going at a constant speed and continue flying at its optimal speed if
it increases its altitude so as to reduce the air density. Next time you’re
on a long-distance flight, you could check whether the pilot increases the
cruising height from, say, 31 000 feet to 39 000 feet by the end of the flight.

### How would a hydrogen plane perform?

We’ve already argued that the efficiency of flight, in terms of energy per
ton-km, is just a simple dimensionless number times g. Changing the
fuel isn’t going to change this fundamental argument. Hydrogen-powered
planes are worth discussing if we’re hoping to reduce climate-changing
emissions. They might also have better range. But don’t expect them to be
radically more energy-efficient.

You can think of dFuel as the distance that the fuel could throw itself if it suddenly converted all its chemical energy to kinetic energy and launched itself on a parabolic trajectory with no air resistance. [To be precise, the distance achieved by the optimal parabola is twice C/g.] This distance is also the vertical height to which the fuel could throw itself if there were no air resistance. Another amusing thing to notice is that the calorific value of a fuel C, which I gave in joules per kilogram, is also a squared-velocity (just as the energy-to-mass ratio E/m in Einstein’s E = mc2 is a squared-velocity, c2): 40 × 106 J per kg is (6000 m/s)2. So one way to think about fat is “fat is 6000 metres per second.” If you want to lose weight by going jogging, 6000 m/s (12 000 mph) is the speed you should aim for in order to lose it all in one giant leap.
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