Free fall and motion under gravity

Falling with only gravity

An object in free fall has only gravity acting on it: no air, no push. It speeds up by $g = 9.8\ \text{m/s}$ every second, whatever its mass.

The equations of motion with a = g

Take down as positive for a drop. From rest:

For a throw straight up, take up as positive and use $a = -g$: the object slows, stops at the top, and falls back.

You should find that the stone on the Moon takes about two and a half times as long. Four times the height only doubles the time, because $t$ grows as $\sqrt h$.

Watch out. Real falling objects meet air resistance. A feather or a parachute is far from free fall; a stone over a short drop is close to it.

A worked example

A stone is dropped from a $45\ \text{m}$ tower. How long does it take and how fast does it land, on the Earth and on the Moon?

1. Earth. $t = \sqrt{\dfrac{2 \times 45\ \text{m}}{9.8\ \text{m/s}^2}} = 3.03\ \text{s}$; $v = 9.8\ \text{m/s}^2 \times 3.03\ \text{s} = 29.7\ \text{m/s}$.
2. Moon. $t = \sqrt{\dfrac{2 \times 45\ \text{m}}{1.6\ \text{m/s}^2}} = 7.5\ \text{s}$; $v = 1.6\ \text{m/s}^2 \times 7.5\ \text{s} = 12\ \text{m/s}$.
3. So on the Moon it falls more slowly and lands more gently.

Practise with new numbers

Try it in Python

The program works out the tower on both worlds.

Remember these

• Free fall: only gravity acts; the acceleration is $g$ for every object.
• From rest: $v = gt$, $h = \tfrac12 gt^2$, $v^2 = 2gh$.
• Thrown up: slows by $g$ each second, stops, falls back.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A ball falls freely for $2\ \text{s}$. How fast is it moving? ($g = 9.8\ \text{m/s}^2$)

2. Is a skydiver with an open parachute in free fall?

3. A stone dropped from rest falls $5\ \text{m}$ in the first second on some planet. What is $g$ there?

4. What is the velocity of a ball at the top of a vertical throw, and what is its acceleration?