Research question
How does varying the drop height (40 cm, 80 cm, 120 cm, 160 cm and 200 cm) affect the ball bounce height?
Introduction
Bouncing balls is an elementary topic in mechanics that demonstrates scientific ideas like energy conservation, momentum, and elasticity of materials. The theoretical correlation between the drop height and the bounce height is usually oversimplified in basic physics courses; however, in practice, there are various complicating factors such as air drag, friction inside the ball (called hysteresis effect), and sound energy losses. It is essential to understand how these inefficiencies operate because of their significance to sports engineering, where the performance of the ball is strictly controlled, as well as other areas in material science and impact mechanics.
The main objective of this investigation is to establish a mathematical relationship between drop height (cm) and bounce height (cm) of tennis ball. This study seeks to answer the research question; How does varying the drop height (40 cm, 80 cm, 120 cm, 160 cm and 200 cm) affect the ball bounce height?
Background information
According to the first law of thermodynamics, energy cannot be created nor destroyed but it can be transformed from one form to the other. When a ball is held at a given height above the ground, it has gravitational potential energy due to its position (Drake 2026). As the ball is released, the potential energy (gravitational) will be converted into kinetic energy (Ke) as the ball accelerates towards the ground. Consider the equation below;
Given that;
- m – mass of the dropping ball
- g – acceleration due to gravity (9.81 m/s²)
- h – release height (m)
As the ball accelerates towards the ground just before the impact;
However, not all the energy will be converted to kinetic energy as some will be converted to sound and heat. As a result, the ball will rebound at a lower level compared to the initial drop.
Aim
The main objective of this exploration is to investigate how varying the drop height (40 cm, 80 cm, 120 cm, 160 cm and 200 cm) affects the ball bounce height. In this exploration the rebound height will be measured using measuring tape.
Hypothesis
Potential energy increases as the height of the ball increases as a result; the converted kinetic energy will also increase with increase in drop height. Based on this information therefore, there is positive correlation between drop height and bounce height. As the height increases, the rebound height also increases. The graph of release height vs bounce height will have a positive gradient confirming a positive correlation between variables.
Variables
Dependent variable
The dependent variable for this investigation is bouncing height of the ball after impact. The bounce height will be measured using same measuring tape.
Independent variable
The independent variable in this experiment is the drop height of the ball (40, 80, 120, 160 and 200 cm). The drop height of the ball will be measured using a measuring tape attached to a vertical surface.
Control variables
| Control variable | Impact on the final results | How the variable will be controlled |
|---|---|---|
| Ball type | Different ball types will lose different amounts of energy inside the ball on impact, and this will have an impact on the bounce height (Britannica 2020). | To have accurate and reliable results, it is important to have same ball (material). This will ensure the same energy loss on impact and thus ensuring data accuracy. |
| Ball size | Different ball size has different masses, radii and internal pressure and this will affect elasticity. Change in elasticity will impact the bounce height of the ball thus impacting the final results. | Same ball with; same mass, radius and pressure will be applied in the entire experiment to ensure data accuracy and consistency. |
| Surface type | The hardness or the elasticity of the surface will impact the amount of energy absorbed by the ground as well as the amount returned to the ball (Courtship 2025). This will in turn affect the rebound height. | To have reliable and accurate data, all the drops will be performed in hard flat surface (concrete). This will ensure accurate and reliable results. |
| Drop technique | Adding initial force introduces initial kinetic energy and this will invalidate the (u = 0) assumption. | To have accurate and reliable results, the ball will be dropped at rest by carefully releasing fingers without pushing. |
| Air resistance | Strong air current/resistance will alter the trajectory of the ball and this will bring the random errors. | To ensure the data is accurate, the experiment will be conducted in a closed environment. |
| Same materials | Different materials and apparatus such as measuring tape and camera have different uncertainty and this will impact the final results. | To ensure data accuracy and consistency, same apparatus will be used in the entire experiment. |
Materials
The table below indicates some of the material used in this investigation;
| Material / Apparatus | Measurement Uncertainty | Usage in the Investigation |
|---|---|---|
| Standard tennis ball | — | Used as the falling object whose bounce height will be measured. |
| Flat concrete surface | — | A flat concrete surface (level and dry). |
| Meter tape | ±0.01 m | Used to measure drop height |
| Smartphone with High-Speed Camera | Frame-rate dependent time resolution | Used to record the motion of the falling ball for frame-by-frame position tracking in Tracker software. |
| Tripod and stand | — | The stand will be used to hold the smartphone steady and also perpendicular to the plane of motion. This will help to prevent camera to shake and ensures consistent framing. |
Method
- Gather all the materials in room with hard concrete surface (do not use carpet). The windows need to be closed to avoid wind and air flow.
- Attach a measuring tape vertically on the wall (mark 0 cm mark) positioned at the level of concrete.
- Mount smart-phone with high-speed camera mounted on a tripod about (2–3) meters. Ensure the camera is positioned perpendicular to plane of motion.
- Adjust the camera to ensure the entire measuring tape from 0 cm–200 cm and the ball motion is visible in the entire experiment.
- Collect one tennis ball with constant mass, radius and material (this will ensure accurate data).
- Position the ball at 40 cm (0.4 m) mark and release the ball without applying any force. (this will ensure the initial velocity is zero)
- Record the bounce height of the ball (cm) and record the data.
- Repeat steps 2–7 four more times to ensure data accuracy and reduced random errors.
- Repeat steps 2–8 by adjusting the release height to (80 cm, 120 cm, 160 cm and 200 cm).
- Record all the data (rebound height in table below).
Risk Assessment
| Hazard | Risk | Control measure |
|---|---|---|
| Dropping ball at height | There is a risk of someone being struck by the ball and this might cause minor injury. | Clear the drop area before each trial. It is also important to stand on the side when releasing the ball. |
| Equipment | There is high risk of equipment damage during, before and after the experiment. | Handle all the equipment with a lot of care to avoid damage. |
Environmental and Ethical Concerns
- Energy use was minimal because only a camera and simple measuring tools were needed, which kept the environmental impact low.
- There were no ethical issues since the experiment did not involve living organisms or personal data.
Raw data
| Drop height (m) | Trial 1 | Trial 2 | Trial 3 | Trial 4 | Trial 5 |
|---|---|---|---|---|---|
| 0.40 | 0.19 | 0.20 | 0.21 | 0.20 | 0.21 |
| 0.80 | 0.35 | 0.35 | 0.36 | 0.36 | 0.37 |
| 1.20 | 0.50 | 0.52 | 0.51 | 0.50 | 0.51 |
| 1.60 | 0.71 | 0.70 | 0.70 | 0.69 | 0.71 |
| 2.00 | 0.90 | 0.90 | 0.91 | 0.90 | 0.90 |
Based on the table above, it can be noted that as the drop height increases, the rebound height also increases.
Sample calculation
Mean
To calculate the mean rebound height, the following formula will be used;
The same approach was used to calculate the average rebound height of the ball.
Uncertainty calculation
To calculate the uncertainty, the following approach will be applied;
To find the uncertainty, consider the following formula;
To find uncertainty when drop height was (40.0 cm)
The same approach was applied to find uncertainty of rebound height. At 2.00 m, half the range (0.005 m) is smaller than the 0.01 m resolution of the readings, so the larger value, ±0.01 m, is used.
| Drop height (m) | Average rebound height (m) | Uncertainty (m) |
|---|---|---|
| 0.40 | 0.20 | ±0.01 |
| 0.80 | 0.36 | ±0.01 |
| 1.20 | 0.51 | ±0.01 |
| 1.60 | 0.70 | ±0.01 |
| 2.00 | 0.90 | ±0.01 |
Analysis
The table above indicates drop height, average rebound height and uncertainty. Based on the table above, it can be noted that as the drop height increases, the average rebound height also increases. When the drop height was 0.40 m, the average rebound height was 0.20 m. As the height increases to 2.00 m, the average rebound height also increases to 0.90 m. This confirms that there is positive correlation between drop height (m) and rebound height (m). The data in the table above was used to develop the following graph;
- Mean of the trials, with its uncertainty
- Line of best fit: y = 0.436x + 0.011, R² = 0.996
Pick a drop height to see its five trials and how its mean and uncertainty were worked out.
Drop height 0.40 ± 0.01 m
- 5 trials
- 0.19, 0.20, 0.21, 0.20, 0.21 m
- Mean
- (0.19 + 0.20 + 0.21 + 0.20 + 0.21) / 5 = 0.202 m, rounded to 0.20 m
- Uncertainty
- (max − min) / 2 = (0.21 − 0.19) / 2 = ± 0.01 m
- Line of best fit
- 0.186 m at 0.40 m; the mean is 0.016 m above the line
| Drop height (m) | Mean rebound height (m) | Uncertainty (m) | Line of best fit (m) |
|---|---|---|---|
| 0.40 | 0.202 | ± 0.01 | 0.186 |
| 0.80 | 0.358 | ± 0.01 | 0.360 |
| 1.20 | 0.508 | ± 0.01 | 0.534 |
| 1.60 | 0.702 | ± 0.01 | 0.709 |
| 2.00 | 0.902 | ± 0.01 | 0.883 |
Based on the graph above, it can be noted that there is positive correlation between drop height and rebound height. As the drop height of the tennis ball increases, the rebound height on the other hand also increases. The line of best fit from the above graph indicates an upward trajectory confirming that there is a positive correlation between the variables. As the drop height increases, more potential energy will be converted into kinetic energy and as such the rebound height increases. The R² value from the graph above is 0.996 (r = 0.998), confirming a strong positive correlation between the variables. The error bars on the graph above do not overlap, showing that each increase in drop height changed the rebound height by more than the spread between trials.
Try the experiment
A simple model of the drop, kept apart from the IA’s data. Choose a drop height, press Drop the ball and see the first bounce the model predicts, next to what the IA measured from that height.
The ball’s lowest point is shown against the ruler. The animation runs at half speed.
Model prediction
0.54 m
First bounce = e² × drop height = 0.67² × 1.20 m = 0.54 m
The ball hits the floor at 4.85 m/s and leaves it at 3.25 m/s, e times as fast.
Measured in this IA
0.508 ± 0.01 m
Mean of the five trials from 1.20 m: 0.50, 0.52, 0.51, 0.50, 0.51 m.
The model gives 0.03 m more than the mean.
e is a parameter of the model, not a new measurement. It starts at 0.67, the value that best fits the IA’s five means with hb = e²hd. The measured ratio hb/hd runs from 0.42 to 0.51 across the five heights, so one constant e is an approximation.
The model assumes:
- The ball is dropped from rest and moves straight down and straight back up.
- The same ball bounces on the same concrete floor every time.
- e is the same at every height.
- Air resistance and the ball’s spin are ignored.
Conclusion
The aim of this investigation is to examine how varying the drop height (40 cm, 80 cm, 120 cm, 160 cm and 200 cm) affects the ball bounce height. The investigation was successful in proving that there is a strong positive correlation between the drop height and the rebound height of a tennis ball. The results revealed that the greatest drop height was 2.00 m with an average rebound height of 0.90 m while that of 0.40 m was 0.20 m. The results of the correlation coefficient test (r = 0.998) showed that there was a strong positive linear relationship, which supported the hypothesis. The explanation is that as the drop height increases, so does the gravitational potential energy which consequently increases the kinetic energy before impact. The rebound height was still less than the original height because energy has been lost through heat, sound and deformation. Overall, the results agree with the predicted correlation between the drop height and the rebound height.
Evaluation
Strength
The exploration was a huge success as the aim of the investigation has been achieved and the hypothesis has been confirmed. The strength of this investigation can be attributed to high number of trials (5). Conducting 5 trials for each drop height increased data reliability and reduced the impact of random errors.
Use of video analysis (Tracker software) – video tracking helped eliminate the effect of human reaction time error, which would have been present if the experiment had been done manually.
Controlled variables like; mass of the ball, surface type and drop technique ensure that the cross-sectional area and surface properties were constant, thus allowing the drop height to be the only variable that changes.
Weaknesses
Although the exploration was a huge success, there are some factors which might have impacted the final results as indicated by the table below;
| Limitation | How It Affected the Results | Suggested Improvement |
|---|---|---|
| Ball rotation during release | If the ball is rotating, the airflow around the ball could change, which would have some effect on the drag. The spin could also have some effect on the accuracy of the tracking. | A mechanical release mechanism, such as a clamp or an electromagnet, can be used. |
| Human error when releasing the ball | Hand release of the ball can include the added force or side motion of the hand. This alters the initial velocity of the ball, and could impact the maximum rebound height, adversely impacting the accuracy and reliability of the results. | The ball will be released by a mechanical release mechanism, such as an electromagnetic release system, or a clamp release system, which will not require extra force to release the ball. This will maintain a ball’s initial velocity of zero in each trial. |
| Assumption of constant air density | Air density was considered constant, although any change in temperature or pressure would cause the drag force to change slightly. | Room temperature can be measured and the real density of the air can be calculated, or the experiment can be carried out in a laboratory environment. |
