Moving Through Air
Why Small Things Fall Slowly and Fast Things Pay Dearly
Two Ways a Fluid Resists
Drop a marble and a grain of pollen and they fall by different rules. Both are held back by the air, but the marble is fighting the air’s inertia, shoving it aside and paying for the momentum it hands over, while the pollen grain is fighting the air’s stickiness, dragging layers of it along by viscosity. The first kind of resistance grows with the square of the speed. The second grows in simple proportion to it.
Which one you meet is decided by a single number, worked out by Osborne Reynolds in 1883: how big and fast the object is, compared with how thick the fluid is. Large and fast means inertia wins and the square law rules. Small and slow means viscosity wins and the linear law rules. A swimmer, a car and a raindrop live in the first world. A bacterium, a dust mote and a fog droplet live in the second, where stopping is instant and coasting is impossible; the Life as Physics page describes what it is like to live there. The Turbulence page follows the same number to the place where smooth flow breaks up.
The square law has a shape everyone has felt. Doubling your speed on a bicycle means four times the force of the air against you, and since you cover ground twice as fast, eight times the power. Wind resistance is the reason a cyclist tucks, the reason a car’s fuel use climbs steeply above motorway speed, and the reason the fastest road cars stop near five hundred kilometres an hour, with every extra kilometre costing more than the last.
Terminal Velocity
A falling object speeds up until the drag on it equals its weight, and then it stops speeding up. That speed is its terminal velocity, and for anything in the square-law world it depends on one ratio: weight against frontal area. A skydiver spread flat reaches about two hundred kilometres an hour; head down, with less area for the same weight, closer to three hundred. A raindrop a few millimetres across arrives at around nine metres a second, slow enough to be harmless. Hail, denser and larger, arrives fast enough to dent cars.
Weight grows with volume, which is length cubed. Area grows with length squared. So shrink an animal and its weight falls faster than the area the air can push on, and its terminal velocity falls with it. The biologist J. B. S. Haldane put it in 1926 as a rule about dropping animals down a mine shaft: the mouse walks away, the rat is killed, a man is broken, a horse splashes. An ant cannot be hurt by a fall from any height, because it never gets going.
Shrink further and the rules change again. Below about a tenth of a millimetre the linear law takes over, and terminal velocity falls with the square of the size. A fog droplet a hundredth of a millimetre across falls at about three millimetres a second, which is why fog hangs. A speck of fine dust settles through a still room over hours. Pollen and bacteria, released into the air, do not so much fall as drift, and that is why the air is full of them.
Shape
In the square-law world the force also depends on shape, through a number called the drag coefficient: how much of the air’s momentum the object actually intercepts. A flat plate facing the wind scores about one point two. A sphere about half that. A modern car about a quarter. A teardrop or a well-designed airship about a twentieth. The teardrop is the shape that lets the air close up smoothly behind it, so that the wake, where most of the drag is made, stays small.
The counterintuitive part is that roughness can help. A golf ball’s dimples trip the thin layer of air on its surface into turbulence, and the turbulent layer clings farther around the ball before letting go, leaving a smaller wake. A dimpled ball flies about twice as far as a smooth one. The Turbulence page tells that story properly, because it is a story about turbulence. Here it is enough to notice that drag is made behind an object, not in front of it, and that the shape of a thing’s tail matters more than the shape of its nose.
Lift, in One Section
Turn a flat plate slightly into the wind and the air is deflected downward as it passes. By Newton’s third law the plate is pushed up, and that push is lift. A wing is a plate shaped to do this smoothly at a small angle, so that the air follows the upper surface instead of tearing away from it. The pressure on top is lower and the pressure beneath is higher, which is the same fact told in the language of pressure rather than momentum. Both are true, and neither is the whole story on its own.
Because lift comes from the same encounter with the air that makes drag, the two scale together. Lift grows with the square of the speed. So a plane needs a minimum speed to fly at all, and needs more of it when heavy or when the air is thin. That is why runways are long, why aircraft take off into the wind, and why a fully loaded airliner on a hot afternoon at a high airport may have to leave cargo behind. The old textbook story about air racing over the longer upper surface to catch up with the air beneath is not how it works. The Turbulence page explains what is wrong with it.
Where This Picture Stops
Everything above treats air as a fluid that gets out of the way and stays the same fluid. Two things break that. Approach the speed of sound and the air can no longer move aside in time; it piles up into a shock, and the drag rises steeply. That wall was a real wall for aircraft in the 1940s, and beyond it the rules are those of compressible flow, with coefficients of their own. Go high enough and the air becomes so thin that the molecules stop behaving as a fluid at all. A space station four hundred kilometres up meets them almost one at a time, and the small drag they produce is still enough to bring it down within a year or two without regular boosts.
And the middle of the range, where turbulence lives, is where the drag coefficients on this page come from measurement rather than derivation. Nobody can compute the drag on a car from the equations of fluid motion alone. The simulations that do it lean on turbulence models tuned to measurements, which is why car makers still own wind tunnels. The rules here are exact in shape and empirical in number. That combination, a clean law with a measured constant inside it, is what most of engineering physics looks like.
