1. The Short Answer: Weight Grows Faster Than Air Resistance
Air resistance (engineers call it drag) is the force you feel when you push something through air. Stick your hand out of a moving car's window and you can feel it shove your hand backward. NASA's Glenn Research Center explains that drag depends on the air's density, on the square of the speed, on the size and shape of the body, and on how it is tilted into the flow. Its size part is a reference area, a flat area that stands in for how much body the air meets [2].
So drag grows with area. Weight is different: it grows with the amount of material, which means volume. Here is the key step. Take an object and make an exact copy of the same shape and material, but with every length twice as big. The area becomes 2 × 2 = 4 times larger. The volume becomes 2 × 2 × 2 = 8 times larger, and so does the weight. The air's push is 4 times bigger, but the weight is 8 times bigger. Compared with its own weight, the air's push has been cut in half.
That is the whole idea. Big things are not magically strong against air. They are just so heavy for their area that the air has a smaller say. This simple count assumes the drag coefficient, a number that bundles the effect of shape and other details, stays the same [2]. We check how well that works later.
2. How Haldane Used This to Explain the Mouse and the Horse
Haldane made the same point with animals. He wrote that the resistance of the air to movement is proportional to the surface of the moving object. If you divide an animal's length, breadth, and height each by ten, its weight falls to a thousandth, but its surface falls only to a hundredth. So, he concluded, for the small animal the resistance to falling is relatively ten times greater compared with the force pulling it down [1].
Haldane was a scientist who liked to explain how size changes the way animals work. In the same essay he made a related point about giants: a person ten times taller, wider, and thicker would weigh a thousand times as much, but the cross-section of the bones would grow only a hundred times [1]. The mouse and the horse are one of several examples.
One caution: air resistance is only part of the story of a fall. How badly an animal is hurt also depends on its bones and body, and Haldane's numbers are a rule of thumb, not a precise measurement.
3. Why Does a Falling Object Stop Speeding Up?
To see what the air really does in a fall, we need the idea of terminal velocity. A falling object feels two forces: weight pulling down and drag pushing up. At first it speeds up. As it goes faster, drag grows. When drag becomes equal to the weight, the forces cancel, the speed stops changing, and the object keeps falling at a steady speed. NASA calls that steady speed the terminal velocity [3].
NASA's page gives the balance as a formula, V = √(2W / (Cd · ρ · A)), where W is the weight, A is the area, Cd is the drag coefficient, and ρ (the Greek letter rho) is the air density. It states what follows from it: an object with a large cross-sectional area falls slower than one with a small area, and a lighter one falls slower than a heavier one of the same size and shape. A large flat plate falls slower than a small ball of the same weight [3].
This also says what Haldane's mouse had going for it. Its weight compared with its area, W/A, is small, so its terminal velocity is low. A horse has a much larger W/A. The page also notes that without air, there would be no drag, and objects would fall at the same rate, as Galileo found [3].
4. How Much Faster Does a Bigger Copy Fall?
We can put numbers on the copies from section 1. If the length is multiplied by L, the area is multiplied by L×L and the weight by L×L×L. So the weight is multiplied by L compared with the area, and the terminal velocity (which depends on the square root of W/A, if Cd stays the same) grows like the square root of L. This last step is our own working from NASA's formula, not a number from the sources.
| Length | Area | Weight | Air's push compared with weight | Terminal velocity |
|---|---|---|---|---|
| 1× | 1 | 1 | 1 | 1× |
| 2× | 4 | 8 | 1/2 | about 1.4× |
| 10× | 100 | 1,000 | 1/10 | about 3.2× |
The last row matches Haldane's wording, run in reverse: make something ten times longer and the air's push, compared with its weight, drops to a tenth [1]. Notice that the fall speed rises much more slowly than the size. Ten times as long gives only about three times the speed.
5. Do Real Raindrops Follow the Rule?
Raindrops are a handy test because they come in many sizes. Japanese Wikipedia, which cites measurements, says drops of 1 mm across fall at about 4 m/s, 2 mm drops at about 6.5 m/s, and 4 mm drops at about 9 m/s (the page lists them by radius; we converted to diameter). It adds that bigger drops do not go faster than about 9 m/s [4]. English Wikipedia gives about 8 m/s for a 3 mm drop, which fits in between [5].
So larger drops do fall faster, as the rule says. But the match is only rough. Going from 2 mm to 4 mm, the size doubles and the speed rises about 1.4 times, close to the square-root rule. Going from 1 mm to 2 mm the speed rises about 1.6 times, a bit more than the rule. And above 4 mm the speed barely changes at all [4]. Our simple count treats Cd as fixed, and a big drop is not a fixed shape. This is our reading of why the numbers drift, not something the sources spell out.
At the other end, cloud droplets with a radius of roughly 1 to 10 micrometers have a terminal velocity under 1 cm/s, so slow that updrafts inside clouds can easily carry them, and they seem to float [4]. For such tiny drops, our squared-speed formula is not a good description. That is a different regime, and a good topic to read about on its own.
6. When Does "Bigger Resists Air Better" Fail?
The rule holds only for copies of the same shape and material. Change either, and you need the real measure, which is weight compared with area and the drag coefficient [3].
- Same size, different weight. A ping-pong ball and a steel ball of the same size have the same area, but the steel ball is far heavier, so it has the larger terminal velocity [3].
- Same weight, different shape. A flat sheet of paper drifts down slowly. Crumple it into a ball and it falls much faster, with the same weight but a smaller area. A parachute uses the same idea: it spreads out to make the area large [3].
- Very small things. As the cloud droplets show, for tiny particles the simple rule gives way to something else [4].
So a more careful way to say it is this: things with more weight for their area resist air better, and growing an object in the same shape makes that ratio bigger.
Test It Yourself with Coffee Filters
Take a flat paper coffee filter and drop it from about as high as you can reach while standing on the floor. Then drop a stack of four filters from the same height as a single filter. If the four are stacked neatly, they have about four times the weight and nearly the same area. If their shape stays about the same as they fall, the formula in section 3 predicts a terminal velocity about twice as high. Is it? Time how long they take to land, and see how close you get. Do not stand on a chair.
A school lab from Vernier does a careful version of this. Students measure the terminal velocity of six stacks of falling coffee filters of different mass and test how well a simple drag model describes them, then improve the model [6]. Haldane's essay is free to read online and is only a few pages [1].