What it shows
A ball that bounces back to a fixed fraction of its previous height gives a geometric sequence of heights. The total distance travelled until it stops is the sum of a geometric series.
The science
If a ball dropped from height h0 rebounds to h1 = r h0, then to h2 = r squared h0 and so on, the heights form a geometric sequence with common ratio r less than 1. The nth bounce reaches h0 r to the power n. A graph of the log of the height against the bounce number is a straight line. The total distance travelled is h0 plus twice the sum of all later heights, which is h0 + 2 h0 r / (1 minus r), since the sum of the series r + r squared + ... is r / (1 minus r). The theory treats the ball as bouncing infinitely, but in practice it stops quickly.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A rubber or tennis ball
- A metre stick fixed upright on a wall
- A hard floor
- A phone to record slow-motion video
- A table for bounce heights
- Graph paper, preferably semi-log
- Calculator
Safety
For this project: Use a ball that is soft enough not to break things and keep it away from windows and glass.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Fix the metre stick vertically, drop the ball from the 100 cm mark and film the fall with the phone beside the stick.
Play the video frame by frame and read the highest point after each bounce for as many bounces as you can see.
Write the heights in a table with the bounce number and calculate the ratio of each height to the one before.
Find the average ratio r. A common mistake is to read the height at the wrong frame, so look for the frame where the ball stops rising.
Plot height against bounce number and also plot the natural log of height against bounce number to show a straight line.
Use the formula h0 r to the power n to predict the sixth bounce height and compare it with your reading.
Calculate the total distance travelled with h0 + 2 h0 r / (1 minus r) and say how near this is to a value that you can estimate from your table.
Repeat for a different ball and on a carpet and compare the ratios r.
What you should see
A nearly constant ratio r between successive heights, perhaps 0.6 to 0.8 for a rubber ball on a hard floor, and a straight line on the log graph. The predicted total distance agrees with the measured data.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is a geometric sequence?
A sequence in which each term is a fixed multiple of the previous one.
2Why does the sum of an infinite series with r less than 1 remain finite?
The terms shrink fast enough, so the total approaches a limit.
3What does r tell us physically?
It is related to how much energy the ball keeps in each bounce, since energy is proportional to height.
4Why does a carpet lower the ratio?
The carpet absorbs more energy at each bounce.
Take it further
Calculate the time between bounces and show that it also forms a geometric sequence, with ratio equal to the square root of r.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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