Moving Through Air
Why Small Things Fall Slowly and Fast Things Pay Dearly
The Air Has to Go Somewhere
Drop a hammer and a feather in a room and the hammer wins by seconds. On the Moon the two land together, so the difference is the air. But what is the air actually doing to the feather?
Anything that moves through air has to push the air in its way aside, and the air pushes back. That push back is drag. The body hands some of its motion to the air, and the air carries it on. Long after a ball has flown by, the air it passed through is still swirling. Push the air downward instead of aside, and the push back can hold a body up, which is how a wing flies.
Counting what the body hands over explains the most important fact about the drag on a ball, a car or a cyclist: it grows roughly with the square of the speed. Go twice as fast and you push aside twice as much air every second, and you have to set each bit of it moving twice as fast. Twice times twice makes four times the drag.
The swirls a body leaves do not last. They break into smaller swirls, and in the end the energy of their motion is shared out among the molecules of the air as a faint warmth. It is the same fate as the energy of motion lost in a crash.
Drag Is Made Behind
The strange thing is that a perfect fluid would hold nothing back at all. Work out how hard a fluid with no stickiness pushes on a body gliding deep inside it at a steady speed. Round a body shaped the same at front and back, the fluid piles up against the front, pushing back, and closes in behind, pushing forward just as hard. The total comes to nothing, and it comes to nothing for every other shape too. It is a paradox, because every thrown ball and every submarine says otherwise.
What the perfect fluid leaves out is stickiness. However thin a fluid is, the layer touching a body clings to it, and a thin skin of slowed air is dragged along with the surface. That thin skin decides where the flow lets go. Round the front of a ball the passing air speeds up and its pressure falls. Toward the back it has to slow down again and push its way back up to a higher pressure. The air in the slowed skin has too little speed left to make that climb. Near the widest point it stops, turns back, and the flow peels away from the surface.
Behind the ball is a wide wake of churning air, and the pressure there stays low. The air presses on the front of the ball harder than on its back, so the ball is held back by the difference. Most of a ball’s drag is made this way, behind it.
So the cure is a tail. Taper the back of a body gently and the air can climb back to a high pressure slowly enough to stay on, and it closes in behind with almost no wake. A teardrop shape, blunt end forward, has about a tenth of the drag of a ball as thick. At ordinary speeds the nose can be round; it is the tail that has to taper.
Roughness can do the same job, oddly. A golf ball’s dimples stir its clinging skin into turbulence, which mixes faster air down into it, and a turbulent skin holds on farther round the back before it lets go. The wake narrows and the drag is roughly halved.
The Price of Speed
Drag that grows as the square of the speed is expensive. The work done against it each second is the drag times the distance covered in that second. At twice the speed that is four times the drag over twice the distance: eight times the power. On the flat at racing speed, about nine tenths of a cyclist’s effort goes into pushing air out of the way.
That is why racing cyclists ride in a line. The rider behind travels in the wake of the one in front, where the air is already moving their way, and needs about a quarter less push. The leader gains a little too, up to about three percent, because the follower fills the low-pressure hollow behind the leader that was holding the leader back. Drag really is made behind. Deep inside a large bunch, a rider meets between a twentieth and a tenth of the drag of riding alone.
Cars pay the same bill. In tests of 74 cars and light trucks, going from 80 to 130 kilometers an hour cut the distance driven on each liter of fuel by about a third. And water is about eight hundred times denser than air, so in water the same shape at the same speed pays roughly eight hundred times as much.
Falling Until the Air Holds You
A falling body speeds up, and the faster it falls the harder the air pushes back. Sooner or later the push equals the body’s weight. From then on nothing speeds it up, and it falls at a steady speed, its terminal speed. The feather in the room reached its own a few centimeters after it was let go.
What sets that speed is mainly weight against the face the air pushes on. A skydiver spread flat falls at about 200 kilometers an hour; head down, with the same weight on a smaller face, at 240 to 290. A sheet of paper crumpled into a ball falls faster than the same sheet held flat.
Raindrops show the push itself. The air presses hardest on the bottom of a falling drop, where it meets the drop head on, and less than still air does at its sides, where it rushes past. A small drop holds itself round. A bigger one is squashed flat underneath, like the top of a hamburger bun, and is never the teardrop it is usually drawn as. The biggest raindrops, about five millimeters across, fall at about nine meters a second.
There is a limit. Past about six millimeters across, the push beats the pull of the water’s surface that holds a drop together. The drop flattens into a pancake, the air blows its middle out into a bag, and the bag bursts into a spray of smaller drops, all within a few hundredths of a second. So rain cannot fall much faster than nine meters a second, and it does not hurt.
Hail is another matter. It is ice, a little less dense than water, so it is no heavier for its size. But it is solid, and nothing tears it apart as it grows. A hailstone as big as a golf ball lands at two to three times the speed of the fastest rain.
Why Small Things Float
Shrink a body and its weight shrinks faster than its face. Weight goes with the volume, which is length times length times length, while the face goes with length times length. So a smaller body is held back more for its weight, and falls more slowly. One biologist pictured a fall down a mine shaft a thousand yards deep: a mouse lands and walks away, while a rat is killed, a man is broken and a horse splashes.
Shrink further and the rule itself changes. For a small enough ball the drag comes from the air’s stickiness alone. It grows with the ball’s width rather than with its face, while the weight still goes with the volume. Halve such a drop and it falls four times more slowly.
A water drop a tenth of a millimeter across falls about a quarter of a meter a second. A fog droplet a hundredth of a millimeter across falls about three millimeters a second, and seems to hang. So a cloud’s droplets fall so slowly that the gentlest rising air holds them up, and those that sink out of its base dry up again. A speck of fine dust takes hours to settle through a still room.
Which rule a body lives under is decided by one number: its size times its speed times the fluid’s density, against how sticky the fluid is. It is called the Reynolds number. A big raindrop scores in the thousands and a skydiver in the millions, and there the square law holds. A fog droplet scores a few thousandths. Below about one, stickiness rules and coasting stops: a bacterium that stops swimming comes to rest in less than the width of an atom.
Where the Air Stops Being a Fluid
Everything so far treats air as a smooth fluid that gets out of the way. It gets out of the way because it is warned. Every push on the air spreads out as a pressure wave at the speed of sound, so the air ahead of a moving body starts to move aside before the body arrives.
At the speed of sound, the warning can no longer get ahead. The air is struck with no notice, all at once, in a thin wall of crowded air called a shock wave. The drag starts to climb steeply a little before, because the air speeding round a body reaches the speed of sound first. For aircraft this was once a real barrier, first broken by a plane with a rocket engine.
Thin air holds a falling body back less, so it lets it fall faster. One skydiver jumped from a balloon 39 kilometers up, where the air is a few hundred times thinner than at the ground. Fifty seconds later he was falling at 1,357.6 kilometers an hour, a quarter faster than sound travels at that height. Then the thickening air slowed him again, as the square law says it should.
Higher still, air stops behaving as a fluid at all. At sea level a molecule of air travels less than a ten-thousandth of a millimeter before it hits another, so to anything we can see, air is a smooth crowd. Four hundred kilometers up, where the air is mostly single atoms of oxygen, each travels kilometers between hits. The International Space Station, about a hundred meters across, meets them as separate blows on its front, each atom striking with no warning passed ahead. That is the simplest picture of air resistance come true. The air there is very thin, but at nearly eight kilometers a second its blows add up. Without a rocket boost every so often, the station would fall back into the atmosphere within one to two years.
The same happens at the small end, in ordinary air. A speck of smoke a few ten-thousandths of a millimeter across is not much bigger than the gaps between those hits, and slips through them, so it falls faster than the rule for small drops allows. This mattered when tiny drops of oil, timed as they fell through air, were used to measure the charge of the electron. The charge came out different for drops of different sizes until the rule was corrected for the gaps.
Between these edges the rules on this page hold, and their shapes are certain. The number in front of the square law is another matter. For a real ball, car or cyclist it still has to be measured, or computed with models of turbulence that are checked against measurements. For a car at highway speed, the churning wake where most of the drag is made is still too fine-grained to calculate outright.
