Free fall under gravity

The same for everyone

Drop a coin and a crumpled sheet of paper together. They land almost together. Drop a coin and a flat sheet: the sheet floats down slowly. Is that because it is lighter? Crumple it, and it falls as fast as the coin. The difference was the air, not the weight.

Without air, every body near the Earth speeds up downwards at the same rate, the acceleration due to gravity:

Every second, a falling body gains $9.8\ \text{m/s}$ of downward velocity. On the Moon, with no air, an astronaut dropped a hammer and a feather together in 1971, and they landed together.

Free fall uses the same equations

Free fall is motion with a steady acceleration, so the three equations of motion work. Take up as positive. Then $a = -g$:

Here $y$ is the displacement from the starting point (positive above it).

At the top. A ball thrown up has $v = 0$ at the top, for an instant. Its acceleration is still $g$, downwards. If the acceleration were zero there, the ball would stay in the air.

Up and down take the same time. The way down is the way up played backwards. A ball thrown up at $u$ reaches the top after $\dfrac{u}{g}$, and comes back to the hand after $\dfrac{2u}{g}$, at the same speed $u$.

Try it

A worked example

A girl on a balcony $20\ \text{m}$ above the ground throws a ball straight up at $15\ \text{m/s}$. It misses the balcony on the way down. Take up as positive and the balcony as the origin.

1. Greatest height. At the top $0 = (15\ \text{m/s})^2 - 2 \times 9.8\ \text{m/s}^2 \times y$, so $y = 11.5\ \text{m}$ above the balcony, $31.5\ \text{m}$ above the ground.
2. When does it land? The ground is at $y = -20\ \text{m}$: $-20\ \text{m} = 15\ \text{m/s} \times t - 4.9\ \text{m/s}^2 \times t^2$, so $4.9\ \text{m/s}^2 \times t^2 - 15\ \text{m/s} \times t - 20\ \text{m} = 0$, and, since $\sqrt{225\ \text{m}^2/\text{s}^2 + 392\ \text{m}^2/\text{s}^2} = 24.8\ \text{m/s}$, $t = \dfrac{15\ \text{m/s} + 24.8\ \text{m/s}}{9.8\ \text{m/s}^2} = 4.07\ \text{s}$. (The other root is negative: a time before the throw.)
3. Landing speed. $v^2 = (15\ \text{m/s})^2 + 2 \times 9.8\ \text{m/s}^2 \times 20\ \text{m} = 617\ \text{m}^2/\text{s}^2$, so $v = 24.8\ \text{m/s}$, downwards.

Watch out. Give every quantity its sign: $u = +15\ \text{m/s}$, $a = -9.8$, and the ground at $y = -20$. Mixing up "the height" with "the displacement" is the most common slip in these questions.

Practise with new numbers

Try it in Python

The program follows the balcony ball every half second until it lands.

Remember these

• Without air, all bodies fall with $g = 9.8\ \text{m/s}^2$, whatever their mass.
• Up positive: $v = u - gt$, $y = ut - \tfrac12 gt^2$, $v^2 = u^2 - 2gy$.
• At the top: $v = 0$, $a = g$ downwards. Up and down take equal times.

Check yourself

1. A stone is dropped from rest. How fast is it falling after $1.2\ \text{s}$, and how far has it fallen?

2. A ball is thrown up at $9.8\ \text{m/s}$. When is it at the top, and when is it back in the hand?

3. A heavy and a light ball are thrown up together at the same speed, with no air. Which goes higher?

4. A student says the acceleration of a ball at the top of its flight is zero, since it stops there. What would happen if that were true?