Free fall

What is it about?

When you let go of an object, it falls faster and faster. Without air resistance, free fall is a uniformly accelerated motion with the gravitational acceleration $g$. In Central Europe $g \approx 9.81\,\tfrac{\text{m}}{\text{s}^2}$. Starting from rest, the laws of free fall are

For a drop height $h$ this gives the fall time and the impact speed

Converting to km/h: An hour has $3600\,\text{s}$ and a kilometre $1000\,\text{m}$. So $1\,\tfrac{\text{m}}{\text{s}} = 3.6\,\tfrac{\text{km}}{\text{h}}$: multiply by $3.6$.

Stroboscope: If a lamp flashes at equal time intervals $\Delta t$, you see the ball at several positions. The gaps grow in the ratio $1 : 3 : 5 : 7 : \ldots$ – each gap is $g \cdot (\Delta t)^2$ larger than the one before.

Mass does not matter: Without air, a heavy ball and a light feather fall equally fast. In air, air resistance slows the feather down strongly: it soon reaches a small terminal velocity and then falls almost uniformly. On the Moon (no air), an astronaut of the Apollo 15 mission dropped a hammer and a feather in 1971 – both reached the ground at the same time.

Other celestial bodies have different values of $g$: Moon $1.62\,\tfrac{\text{m}}{\text{s}^2}$, Mars $3.71\,\tfrac{\text{m}}{\text{s}^2}$, Jupiter $24.79\,\tfrac{\text{m}}{\text{s}^2}$.

Reaction time: If you catch a falling ruler, the distance $s$ it fell gives your reaction time: $t = \sqrt{\frac{2s}{g}}$.

Try it

1. Press Drop and look at the stroboscopic picture. Read the gaps on the blue bar: $1 : 3 : 5 : 7 : 9$.
2. Drag the ball in the drop tower (or the point in the distance–time graph) and watch how time and speed change. The triangle in the velocity–time graph shows the slope $g$, the area under the line shows the distance fallen.
3. Repeat the experiment on the Moon and on Jupiter. On the Moon
4. Turn the feather tube over – once with air, once in a vacuum. Tube with air · Tube in a vacuum
5. Measure your reaction time: press Start and catch the ruler (tap, click or space bar). Do at least five attempts. Reaction test

Tasks

A ball falls from a height of $20\,\text{m}$ (no air resistance). How long does the fall take, and how fast is the ball at the bottom?

The stroboscope flashes every $0.4\,\text{s}$. Calculate the gaps between the first five images and their ratio.

How long does the ball take to fall $20\,\text{m}$ on the Moon? How many times longer is that than on Earth?

Task 4: Depth of a well

A stone falls into a well and hits the water after $2\,\text{s}$. How deep is the well (ignore air resistance and the travel time of the sound)?

Task 5: Reaction time and reaction distance

Someone catches the ruler after $18\,\text{cm}$. What is the reaction time? How far does a car travelling at $50\,\tfrac{\text{km}}{\text{h}}$ move in this time before the driver even starts to brake?

Task 6: Feather and ball

Explain why the feather takes much longer in the tube with air, but falls just as fast as the ball in the evacuated tube. With air · Vacuum

An apple falls from a tree at a height of $5\,\text{m}$. How fast is it when it hits the ground? Give the speed in $\tfrac{\text{m}}{\text{s}}$ and in $\tfrac{\text{km}}{\text{h}}$.

Ideas for the classroom

• Use: Uniformly accelerated motion and the laws of free fall (grades 9/10), historical link to Galileo, application in traffic (reaction time). Each of the three experiments can be opened directly with a link containing mode=….
• Model: The drop tower ignores air resistance. In the tube, air resistance is modelled as proportional to $v^2$, with terminal velocities of about $75\,\tfrac{\text{m}}{\text{s}}$ (steel ball, diameter $3\,\text{cm}$) and $0.6\,\tfrac{\text{m}}{\text{s}}$ (feather) – these are approximate values. The swaying of the feather is only hinted at. Jupiter has no solid surface; the value $24.79\,\tfrac{\text{m}}{\text{s}^2}$ applies at the height of the cloud tops – the drop tower there is a thought experiment.
• Reaction test: The ruler falls after a random waiting time (1.2 to 3.5 s). Screen and input devices add a delay of a few hundredths of a second, so the measured times are usually a little longer than with a real ruler. Comparing with a real ruler (pair work) is a good idea – with the ms scale switched on, the ruler becomes a “reaction time ruler”.
• Common stumbling blocks:
  • “Heavy objects fall faster.” – Only air resistance makes a difference.
  • “In equal times the object falls equal distances.” – The gaps in the stroboscopic picture keep growing, each time by the same amount $g \cdot (\Delta t)^2$.
  • “There is no gravity on the Moon.” – It is only about one sixth as strong.
  • “$g$ is a speed.” – $g$ tells you by how many $\tfrac{\text{m}}{\text{s}}$ the speed increases every second.