Why everything falls at the same rate

TL;DR:

• Gravity pulls harder on a heavier object, but a heavier object is also harder to accelerate. The two effects cancel, so a = g for all masses.

• Without air, a hammer and a feather hit the ground together. On Earth g ≈ 9.8 m/s²; on the Moon g ≈ 1.62 m/s².

• Feathers fall slowly in air because of air resistance, not because they are light.

Everyday experience says heavy things fall faster. A hammer drops straight down and a feather drifts. Yet in a vacuum, the two land at the same moment. The reason is in Newton's second law, and it comes down to one cancellation.

The cancellation in F = ma

Near Earth's surface, the gravitational force on an object of mass m is its weight, F = mg. Newton's second law says the acceleration is the force divided by the mass: a = F / m.

Substitute: a = mg / m = g. The mass is on top and on the bottom, so it cancels. Gravity pulls ten times harder on a ten times heavier object, and that object has ten times the inertia, so the acceleration is the same. The value g is about 9.8 m/s² on Earth.

If you ever get a final answer for a free-fall problem that depends on the mass, look for a slip. Mass shows up in the force, but it drops out of the motion.

Worked example: a hammer and a feather in a vacuum

Say a 1.0 kg hammer and a 3.0 g (0.0030 kg) feather are released together from 1.5 m in a vacuum chamber. Compare their motion.

1. Hammer force: F = 1.0 × 9.8 = 9.8 N. Acceleration: 9.8 / 1.0 = 9.8 m/s².

2. Feather force: F = 0.0030 × 9.8 = 0.0294 N. Acceleration: 0.0294 / 0.0030 = 9.8 m/s².

3. Fall time from h = ½gt²: t = √(2h/g) = √(2 × 1.5 / 9.8) = √0.306 ≈ 0.55 s.

4. Speed on landing: v = gt = 9.8 × 0.553 ≈ 5.4 m/s.

The hammer feels a force more than 300 times bigger, since 9.8 ÷ 0.0294 ≈ 333, yet both accelerate at 9.8 m/s² and take 0.55 s. Check units: √(m ÷ (m/s²)) = √(s²) = s.

Now repeat on the Moon, where g ≈ 1.62 m/s². The fall time is t = √(2 × 1.5 / 1.62) = √1.85 ≈ 1.36 s. The time is longer because the pull is weaker, but still the same for both objects.

What air does to the story

In air, there is a second force upward: air resistance, or drag. It grows with speed and depends on shape and size, not mass. A feather has a large surface for its tiny weight, so drag soon balances its weight, and it settles at a slow steady terminal velocity.

A hammer has far more weight for the same drag, so it reaches a much higher speed before drag catches up. So the difference you see in air is a difference in drag compared with weight. A crumpled sheet of paper falls faster than a flat sheet of the same mass, because you changed only the drag.

The Moon demonstration

In 1971, during the Apollo 15 mission, astronaut David Scott held a hammer and a falcon feather at the same height on the Moon and let go. With no atmosphere, both reached the ground together, on camera. It is the same experiment as the chamber one above, with the vacuum supplied by the Moon.

There is a deeper reason behind this. The mass in F = mg measures how strongly gravity grabs an object. The mass in F = ma measures how much an object resists being accelerated. These are two different jobs for the same property, and experiments show that they match extremely closely. That match is the starting point of Einstein's general relativity.

Galileo is usually credited with the first careful arguments that fall speed does not depend on weight. His reasoning was a thought experiment: tie a heavy and a light stone together. If heavier fell faster, the light one would drag on the heavy one and slow the pair, yet the pair is heavier than either alone and should fall faster. That contradiction suggests they must all fall at one rate.

How to check your answer

• Mass check. Your acceleration should not contain m. If it does, find the slip.

• Round trip. Put t = 0.553 s back into ½gt² = ½ × 9.8 × 0.306 = 1.5 m.

• Units. Metres and seconds throughout; g in m/s².

• Reasonableness. About half a second to fall 1.5 m, roughly a tall person's head height, matches experience.

Drop tests: If you try this at home, drop soft objects from a low height onto a floor, never from windows or balconies, and never above people or pets. Heavy or hard things can injure someone or damage property.

Mistakes that cost marks

• Saying heavier objects fall faster in a vacuum.

• Confusing weight (a force, in newtons) with mass (in kilograms).

• Forgetting that in air, drag, not weight, explains the feather.

• Using g = 9.8 for the Moon or another planet.

• Forgetting to square the time or to take the root in h = ½gt².

Where you will use it next

The same idea of falling underpins orbits: an orbiting object is falling, but moving sideways fast enough to keep missing the ground. For another system where force and mass set the motion, see springs and simple harmonic motion.

Frequently asked questions

Does a bowling ball hit the ground before a tennis ball in air?

Only by a tiny amount for short drops, because drag is small compared with both weights. The lighter ball is slowed slightly more.

Is g the same everywhere on Earth?

It varies a little with latitude and altitude, but 9.8 m/s² is a good value for ordinary problems.