Free Fall: Why a Hammer and a Feather Fall Together

What people believed

For almost two thousand years people trusted Aristotle, who taught in the fourth century BC that heavier things fall faster. It sounds sensible: a stone drops quickly, a leaf floats down. In the animation a heavy ball and a light ball start together, and the old claim seems to hold. But a claim is not a measurement. Galileo, working around 1600, tested falling and rolling bodies and found that heavy and light ones keep almost the same pace. A famous story says he dropped balls from the Leaning Tower of Pisa. It is probably a legend; his careful tests used balls rolling down gentle ramps.

Air is the villain

Drop a feather and a coin and the coin wins easily, but the reason is not weight. Air pushes against anything moving through it, and it pushes much harder on a wide, light feather than on a compact coin. Take the air away, for example by pumping it out of a long glass tube, and both fall together. Without air, gravity gives every object the same acceleration, called g, whatever its mass. At Earth's surface g = 9.81 m/s²: each second of falling adds 9.81 m/s to the speed. In the sandbox, switch the air on and off and compare the feather's fall.

A hammer and a feather on the Moon

On 2 August 1971, Apollo 15 commander David Scott stood on the Moon with a hammer in one hand and a falcon feather in the other, and let go of both at once. The Moon has almost no air, so nothing slowed the feather, and the cameras showed them reaching the dust together. Moon gravity is g = 1.62 m/s², about six times weaker than ours, so the fall looks slow and dreamy. From 1.5 m, roughly shoulder height, it lasts t = √(2 × 1.5 / 1.62) ≈ 1.36 s. On Earth the same drop takes √(2 × 1.5 / 9.81) ≈ 0.55 s.

Distance grows with time squared

Because the speed keeps rising, the distance does not grow in step with the time; it grows faster. The rule is s = g·t² / 2, where s is the distance fallen in metres, g is the pull of gravity in m/s² and t is the time in seconds. On Earth, after 1 s a body has fallen 9.81 × 1 / 2 ≈ 4.9 m, after 2 s it has fallen 9.81 × 4 / 2 ≈ 19.6 m, and after 3 s about 44.1 m. The dots in the animation are placed at equal time steps, and they spread out more and more. The speed after t seconds is v = g·t, so after 3 s it is about 29.4 m/s, or 106 km/h.

Falling from a roof

You can turn the rule around to find how long a fall lasts. Solve s = g·t² / 2 for t and you get t = √(2h / g), where h is the height. A 20 m drop, roughly a six-storey building, takes t = √(40 / 9.81) ≈ 2.02 s, and the body hits the ground at v = g·t ≈ 19.8 m/s, about 71 km/h. Notice that mass appears nowhere in these formulas. That is the whole point of the chapter. Choose a height in the sandbox and compare the readout with your own calculation.

Weight on other worlds

Gravity is not the same everywhere. On the Moon g = 1.62 m/s², on Mars g = 3.71 m/s², and at the cloud tops of Jupiter g = 24.79 m/s². Use the formula for the 20 m drop: it takes √(40 / 1.62) ≈ 4.97 s on the Moon, √(40 / 3.71) ≈ 3.28 s on Mars and √(40 / 24.79) ≈ 1.27 s on Jupiter. An object's mass is the same everywhere, but its weight, the pull it feels, changes with g. A 60 kg person weighs 60 × 9.81 ≈ 589 N on Earth and only 60 × 1.62 ≈ 97 N on the Moon. Try each world in the sandbox.

Free fall and gravity