Bouncing Ball on Two Floors
A student drops a squash ball onto a concrete floor from heights of 20, 40, 60 and 80 cm and records the height of the first bounce: 11, 21, 32 and 41 cm. Using this pattern she predicts a bounce of about 51 cm from a 100 cm drop and about 82 cm from a 160 cm drop. The 100 cm test matches her prediction, so she moves to the gymnasium for the 160 cm drop. There, the ball bounces only 55 cm, and she concludes that the ball behaves differently when dropped from greater heights.
1. Is the prediction for the 100 cm drop reasonable? Justify your answer using the data, and state what assumption it relies on.
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Yes, it is reasonable. Look at how the bounce increases with each 20 cm step in drop height:
| Drop (cm) | Bounce (cm) | Increase per 20 cm |
|---|---|---|
| 20 | 11 | - |
| 40 | 21 | +10 |
| 60 | 32 | +11 |
| 80 | 41 | +9 |
The increase is roughly constant at about 10 cm of bounce per 20 cm of drop, i.e. bounce ≈ 0.5 × drop height. Continuing that pattern gives 41 + 10 ≈ 51 cm for a 100 cm drop.
The assumption is that the same relationship continues between 80 and 100 cm - that the ball's behaviour does not change in that range. Since 100 cm is only a small step beyond the highest measured drop, that assumption is reasonable here (and the test confirmed it).
2. The prediction of an 82 cm bounce from 160 cm is much less secure than the 100 cm prediction. Explain why, even if the gymnasium result is ignored.
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160 cm is double the highest drop actually tested. The pattern is an extrapolation, not an interpolation, and simple linear patterns often break down when pushed far beyond the data.
Reasons to expect a change:
- The ball deforms more on harder impacts, and more energy is lost to internal friction, so bounce height per cm of drop tends to fall off at high speeds.
- Air resistance losses grow as speed increases, again reducing the bounce.
- A squash ball has a squash-ball rating for a reason: its bounce is not guaranteed to scale indefinitely.
So '82 cm' should be presented as a rough guess to be tested, not a prediction she can rely on. A sensible next step is to measure 100 cm and 120 cm before attempting 160 cm, so the extrapolation is shorter.
3. At 160 cm on the gym floor the ball bounced 55 cm. The student concludes that the ball behaves differently at greater heights. Evaluate this conclusion and describe an observation that would test it.
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The conclusion is not justified because two things changed at once: the drop height and the floor. Concrete is hard and returns energy efficiently; a gymnasium floor is usually timber on battens or a sprung synthetic surface, which absorbs impact energy. A lower bounce on a softer floor is expected at any height, so height cannot be blamed.
A test that separates the two variables:
- Drop the ball from 160 cm onto the same concrete floor as before. If it bounces near 80 cm, height is not the cause.
- Drop the ball from 80 cm onto the gym floor. If it bounces well below the 41 cm recorded on concrete (say 28 cm), the floor is the cause of the low bounce.
If both checks come out as above, the correct conclusion is that the gym floor absorbs more energy than concrete, not that the ball behaves differently at greater heights.