Free fall

Dropping things

Drop a stone and a coin together: they land together. Without air, every object falls with the same steady acceleration, called $g$:

So each second, a falling object goes $9.8\ \text{m/s}$ faster. A feather falls slowly only because air pushes on it.

Dropped from rest

With $u = 0$ and $a = g$, the equations of motion become

Thrown upward

A ball thrown up slows by $9.8\ \text{m/s}$ every second until it stops at the top, then falls back. Going up, $a = -g$.

At the top $v = 0$, so the greatest height is $h = \dfrac{u^2}{2g}$, reached after $t = \dfrac{u}{g}$. It takes just as long to come back down.

You should find that doubling the throwing speed makes the ball go four times as high but stay in the air only twice as long.

Watch out. At the top the velocity is zero, but the acceleration is still $g$ downward. That is why the ball does not stay there.

A worked example

A ball is thrown straight up at $14.7\ \text{m/s}$. Find the time to the top, the greatest height and when it comes back.

1. Time to the top. $\dfrac{14.7\ \text{m/s}}{9.8\ \text{m/s}^2} = 1.5\ \text{s}$.
2. Greatest height. $\dfrac{(14.7\ \text{m/s})^2}{2 \times 9.8\ \text{m/s}^2} = 11.0\ \text{m}$.
3. Back down. Another $1.5\ \text{s}$: in your hand after $3.0\ \text{s}$.

Practise with new numbers

Try it in Python

The program works out the ball of the example.

Remember these

• Without air, everything falls with $g = 9.8\ \text{m/s}^2$.
• Dropped: $v = gt$, $h = \tfrac12 gt^2$.
• Thrown up at $u$: greatest height $\dfrac{u^2}{2g}$, time to the top $\dfrac{u}{g}$.

Check yourself

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

1. How fast is a dropped stone going after $3\ \text{s}$?

2. How far has it fallen in those $3\ \text{s}$?

3. A hammer and a feather are dropped together on the Moon. Which lands first?

4. What is the acceleration of a ball at the very top of its flight?