The air taxTwice the speed, four times the shove, eight times the horsepower. This one law explains your gas mileage, the shape of a record car, and why a bicycle racer hides behind another one.
Air weighs somethin'. About 1.2 kilograms a cubic metre at sea level, which sounds like nothin' until you work out how many cubic metres a truck shoulders aside in an hour at seventy — better than a hundred and fifty thousand of them, near two hundred tonnes of air, shoved.
The force it takes goes with the square of the speed:
F = ½ · ρ · v² · Cd · A
ρ is how heavy the air is, v is how fast you are goin', A is how big a hole you are punchin', and Cd is a number for how ugly the shape is — about 0.40 for a pickup, 0.24 for a slick sedan, near 0.04 for a shape drawn for the purpose[33].
But the power — the horsepower, the fuel, the bill — goes with the cube, because power is force times speed. Shoving four times as hard while going twice as fast is eight times the work per second:
| Speed | Air shoved aside | Horsepower, air only | Against 30 mph |
|---|---|---|---|
| 30 mph | 132 N | 2.4 hp | 1× |
| 60 mph | 529 N | 19.0 hp | 8× |
| 80 mph | 940 N | 45.1 hp | 19× |
Doubling thirty to sixty costs exactly 8 times the power. Going from sixty to eighty — a third again as fast — costs 2.4 times. That is the whole story of your gas gauge on the interstate, and the EPA puts the same thing in money: past about fifty, every five miles an hour you add costs you like paying another fifteen to twenty cents a gallon[32].
What the cube law makes people do
Draft. Sit in the hole somebody else already punched and your own bill drops by a third or better. Ducks do it, bicycle racers do it, and stock cars do it at two hundred miles an hour with a foot of air between them.
Get skinny, not slick. The frontal area A is a plain multiplier and the cheapest thing to cut. A record car is needle-thin because every square metre you delete is a square metre you never pay for again.
Go where the air is thin. ρ falls off with height. At sixty thousand feet there is about a tenth the air, so a tenth the drag at the same speed — which is exactly why the SR-71 cruised up where the sky goes dark, and why nothing fast flies low for long.
Quit the air entirely. Above a hundred kilometres or so there is almost nothing left to shove, and that is where the numbers on the ladder stop being thousands and start being tens of thousands. Which is the next page.
The bill you cannot cut
Rolling resistance — tyres flexing — is roughly flat with speed, so it dominates down low and disappears into the noise up high. Somewhere around fifty miles an hour in an ordinary car the air takes over and never gives it back. Below that, a heavy car costs you. Above it, a wide one does.