What if there were no air resistance?

Drop a ball and a sheet of paper. The ball lands first.

Is that because the ball is heavy? Or is something else slowing the paper down?

Hold a ball in one hand and a flat sheet of paper in the other, and let go together. The ball thumps down, while the paper drifts and flutters and lands much later.

It's easy to think heavy things simply fall faster. But there is something else in the way: the air. So what would happen if there were no air at all? In this lab you can switch air resistance off and find out.

Everyday experience suggests that heavy objects fall faster than light ones. Galileo concluded instead that all objects released together fall at the same rate, whatever their mass, once air resistance is out of the way.

This lab tests that idea with a model: a light, cupped piece of paper (10 g, presenting about 0.03 m² to the air) is dropped from 10 m with drag switched on, then off. How much of its slow fall is due to air?

Air is all around us. When things fall, they push through it.

The air pushes back. Light, wide things like paper get pushed the most, so they float down slowly.

Air is made of countless tiny particles. A falling object has to shove them out of the way, and they push back. That push is called air resistance, or drag. It grows bigger the faster something moves.

Our paper cup weighs only 10 grams but is about 20 centimeters across, so drag soon balances its weight. In the model, it stops speeding up at about 2 meters per second and takes about 5 seconds to fall 10 meters.

Drag on a falling object is modeled as F = ½ρC_dAv², with air density ρ = 1.225 kg/m³. As speed rises, drag grows until it equals the weight mg; the object then falls at its terminal speed v_t = √(2mg/(ρC_dA)).

For the paper cup (m = 0.01 kg, C_d = 1.3, A = 0.03 m²) the model gives v_t ≈ 2.0 m/s. It gets close to that speed almost at once, then drifts down the rest of the way, landing after about 5.1 s (a steady 2.03 m/s alone would take 4.9 s).

Olly takes away all the air. Now he drops the paper again.

Whoosh! It falls just like a stone. It lands in about one and a half seconds.

Now switch air resistance off. Without air, there's nothing to push back on the paper cup, so only gravity acts on it. It speeds up all the way down, just like a ball or a stone.

In the model it now lands in about 1.4 seconds instead of 5, hitting the ground at about 14 meters per second. Try changing the mass with the air off: heavy or light, the landing time stays exactly the same.

With drag removed, the paper cup is in free fall: a = g for any mass, so t = √(2h/g) ≈ 1.43 s and v = √(2gh) ≈ 14.0 m/s from 10 m. The terminal speed no longer exists.

For comparison, a football (0.45 kg, C_d 0.47, A 0.038 m²) dropped from 10 m in air lands after about 1.49 s at 12.5 m/s in the model: drag matters much less for dense, compact objects over short drops.

Gravity pulls the paper down. The air pushes it back up a little.

With no air, nothing pushes back. So the paper falls as fast as anything else.

Two forces act on the falling paper: gravity pulling down and air pushing up. For a light, wide object, the air's push quickly grows as big as gravity's pull, so the paper stops speeding up.

Gravity pulls harder on heavier things, but heavier things are also harder to speed up, so without air everything falls together. Our model treats the paper as a rigid shape that doesn't flutter or tumble, so it's a simplified picture of real paper.

Gravity's force mg is proportional to mass, so the acceleration it produces, g, is independent of mass. Drag, by contrast, depends on shape and size, not mass, so its deceleration ρC_dAv²/2m is largest for light objects with a large area. That is why mass matters only when air is present.

Limits: the model assumes a constant drag coefficient, constant air density, a rigid object with no spin, and no wind. Real paper flutters and tumbles, which a constant C_d only approximates.

The Moon has almost no air. An astronaut once dropped a hammer and a feather there.

They landed at the same time!

On the Moon there's almost no air, so it's a natural test lab. At the end of a moonwalk in 1971, Apollo 15 commander David Scott held up a hammer and a feather and let go of both together. With no air to slow the feather, they hit the ground at the same moment.

On Earth, skydivers use air resistance. A skydiver lying flat, belly down, catches more air than one falling feet first, so the air's push balances their weight at a lower speed.

During Apollo 15 in 1971, David Scott dropped a geological hammer and a falcon feather on the Moon; in the near-vacuum they fell at the same rate, as Galileo had concluded centuries earlier (NASA).

Skydivers exploit drag: OpenStax lists C_d ≈ 1.0 for a horizontal skydiver against 0.70 feet first, and a larger C_d·A lowers the terminal speed √(2mg/(ρC_dA)). NASA's terminal-velocity materials show how mass, area and air density each set that speed.