Newton's Laws · Grade 11

Why do a heavy body and a light one fall at the same rate?

Because the same mass appears twice and cancels. The gravitational force on a body is its mass times the gravitational field strength, and the acceleration that force produces is the force divided by that same mass. The mass cancels, and what is left is the field strength alone — the same number for every body, whatever its weight.

Learning objectives

The cancellation, in one line

The gravitational force on a body is its mass times g. Newton's second law says acceleration is force divided by mass. Substitute the first into the second: the mass appears on the top and the bottom and cancels.

The result is that the acceleration equals g, with no trace of the mass in it. A one-kilogram body and a one-tonne body accelerate at exactly the same rate.

This is not a coincidence but a deep statement: the same mass that decides how hard the Earth pulls is also the mass that decides how much the body resists acceleration. The two grow together and so they cancel.

Mass and weight

Mass is measured in kilograms, is a property of the body, and does not change when you move it. Weight is the force a planet exerts on the body, is measured in newtons, and does change with location.

A 60 kg person weighs about 600 N on Earth and about 100 N on the Moon, because the field strength there is roughly a sixth. Their mass has not changed at all.

So a spring balance really measures weight, and a smaller number on the Moon does not mean matter has been lost. This is the distinction that costs the most marks in exams.

A feather and a hammer

In a classroom a feather falls slowly, and the law appears to be wrong. It is not — there is simply another force on the feather that was left out of the account: air resistance.

Air resistance depends on surface area and speed, not on mass, which is why it matters enormously for a feather and hardly at all for a hammer. The net force on the feather is no longer its weight alone.

There is no air on the Moon. An Apollo 15 astronaut dropped a hammer and a feather together there and they landed at the same moment — an experiment showing the prediction is right the moment you remove the force that was hiding it.

Worked examples

  1. What is the weight of a 5 kg body on Earth and on the Moon?

    1. Weight is mass times gravitational field strength
    2. On Earth, 5 times 10
    3. On the Moon the field strength is about a sixth, roughly 1.6

    Answer: About 50 N on Earth, about 8 N on the Moon. The mass is 5 kg in both

  2. A body falls from rest. What is its speed after 3 s, and how far has it fallen?

    1. The speed is acceleration times time: 10 times 3
    2. The distance is half times acceleration times time squared
    3. Half times 10 times 9

    Answer: 30 m/s, having fallen 45 m

  3. Why does a feather fall slowly in a classroom but not on the Moon?

    1. In a classroom air resistance also acts on the feather
    2. That resistance depends on surface area, not mass
    3. On the Moon there is no air and no extra force

    Answer: Because of an extra force that is not gravity, and not because of its mass

Common mistakes

Saying a heavier body falls faster
A larger force does act on it, but it also resists acceleration by exactly the same factor. The two cancel and the same acceleration is left.
Writing mass in newtons
The newton is a unit of force, and therefore of weight. Mass is in kilograms and is the same on Earth, on the Moon and in space.
Concluding from the feather that the law is wrong
The feather demonstrates an extra force, not a broken law. Remove the air and both bodies land together, as was actually done on the Moon.

What to remember

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