Drop anything and, if air resistance is small, it falls with the same constant acceleration. From Mass and Weight, the gravitational force on a mass $m$ is $mg$; Newton's second law gives its acceleration as that force divided by the same mass.
Key idea
In free fall every object accelerates downward at $g = 9.8\ \text{m/s}^2$, whatever its mass. With up as positive, $a = -9.8\ \text{m/s}^2$, and the constant-acceleration equations apply.
Why it works
A heavier object is pulled harder, but it also has more inertia to overcome, in exactly the same proportion: $a = \dfrac{F}{m} = \dfrac{mg}{m} = g$. The mass cancels. In 1971, Apollo 15 astronaut David Scott dropped a hammer and a falcon feather on the airless Moon, and they landed together.
| Time (s) | Speed (m/s) | Distance fallen (m) |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 9.8 | 4.9 |
| 2 | 19.6 | 19.6 |
| 3 | 29.4 | 44.1 |
Worked examples
Worked example: Time to fall
A stone is dropped from a bridge $30\ \text{m}$ above the water. How long does it fall?
- From rest, $h = \tfrac12 g t^2$, so $t = \sqrt{\dfrac{2h}{g}} = \sqrt{\dfrac{60}{9.8}}$.
- $t \approx 2.47\ \text{s}$.
Answer: about $2.47\ \text{s}$
Worked example: Signs with up positive
Take up as $+y$. A ball is dropped from rest. Find its velocity and displacement after $2\ \text{s}$.
- $v = 0 + (-9.8)(2) = -19.6\ \text{m/s}$ (moving down).
- $$\Delta y = \tfrac12(-9.8)(2)^2 = -19.6\ \text{m}$$ (below the start).
Answer: $v = -19.6\ \text{m/s}$, $\Delta y = -19.6\ \text{m}$
Common mistake
Thinking heavier objects fall faster. In everyday life a feather drifts because air resistance matters for it, not because it is light. Without air (or when air resistance is small, as for two steel balls), objects of any mass fall together. Check: $g$ never has a mass in it.
Try it in the Science Lab: the Second Law page lets you change the mass and the force and watch the acceleration.