How do different heights affect the bounce numbers of a table tennis ball after it is dropped?

Introduction

When a ball is raised to a height above ground level, it gains potential energy. Releasing this ball will transform the potential energy into kinetic energy. Contacting the ground creates an impulse causing the ball to bounce back up. By simply referring to this theory as well as intuition, there must be a relationship between the height released and the number of bounces by a ball.

Research Question:How do different heights affect the number of bounces of a table tennis ball after it

is dropped until it comes to rest?

HypothesisIf the table tennis ball is released at a higher height, higher speed leads to stronger

impulse, then the more bounces a ball will have before coming to rest.

Independent variable: HeightThe height will vary as 0.5m, 0.9m, 1.3m, and 1.7m, increasing by 0.4m each time. Under each variation there will be five measurements taken to reduce random error.

Dependent variable: Number of bounces There will be 2 people counting the number of bounces and the final result for each trial will be the average of the two numbers.

Controlled variables: Table Tennis Balls Instead of using different types of balls, table tennis balls are used exclusively throughout the experiment Bounciness To further control the experiment, all the table tennis balls are the same brand (Yaping), and also the same model (AS7); they are also brand new to ensure the table tennis balls would not potentially lose its bounce from time. MethodTo ensure no extra downward force is exerted on the table tennis ball when dropped, the ball is held by a clamp from the horizontal sides. By only squeezing the clamp to release the ball ensures no downward force is provided. TemperatureKnowing temperature will affect the hardness of the plastic, the entire experiment is carried inside the house with a constant temperature of 22℃, according to the value on

the central heating system. EnvironmentThe experiment takes place in a garage, thus the balls are all bounces up from a concrete floor. The doors and windows are closed, so there are no extra air resistance affecting the experiment. The reason this is taken into account is because table tennis balls are very light, and is easily affected by wind. CountersThere are two counters recording the data. The same two people are helping throughout the experiment. Final data will be taken from averaging the two results.

Apparatus- Measuring tape- Six Yuping brand Model AS7 Table Tennis Balls- Clamp

Figure 1: And illustration of the method devised for this experiment.

Procedure1. Locate the designated height with a meter tape.2. Hold the bouncing ball with a clamp on the horizontal side of the ball. 3. Release and the counters should start counting. 4. Note: the order of the heights should be randomized in order to avoid fatigue as well

as pre-cognitive assumptions by the counters after 2-3 trials of one variation. For example, one counter may start to assume the data according to prior trials instead of focusing recording the present one.

5. Repeat the steps until all 25 trials are finished.

Raw Data Table

Released Height (m) ±0.05m Number of bounces by Counter 1

¿)Number of bounces by Counter

2¿)

0.50m 8 10

0.50m 10 9

0.50m 9 7

0.50m 8 9

0.50m 10 8

0.90m 12 11

0.90m 14 13

0.90m 12 10

0.90m 14 11

0.90m 13 12

1.30m 15 16

1.30m 16 15

1.30m 14 15

1.30m 15 16

1.30m 16 15

1.70m 11 12

1.70m 10 13

1.70m 13 12

1.70m 12 11

1.70m 13 14

2.10m 11 12

2.10m 12 9

2.10m 10 11

2.10m 11 12

2.10m 10 11

Table 1: Quantitative data for the bouncing table tennis ball

- Uncertainty in height due to the possibility of fluctuation when performing the experiment, because there has to be enough empty space for the ball to bounce. The ball is slowly moved away from the meter tape to let drop, creating uncertainty.

- Uncertainty in bounces due to human limitation. Even though I have used two counters to decrease the random error, however, there are still likely to be error.

-Qualitative Data- The sound of the first bounce becomes different after reaching 1.7m. It made more

like a breaking sound, while the other bounces are more like a normal, light table tennis ball sound.

Processed Data

Released Height (m) ±0.05m Average bounces ¿)

0.50m 9

0.50m 9.5

0.50m 8

0.50m 8.5

0.50m 9

0.90m 11.5

0.90m 13.5

0.90m 11

0.90m 12.5

0.90m 12.5

1.30m 15.5

1.30m 15.5

1.30m 14.5

1.30m 15.5

1.30m 15.5

1.70m 11.5

1.70m 11.5

1.70m 12.5

1.70m 11.5

1.70m 13.5

2.10m 11.5

2.10m 11.5

2.10m 10.5

2.10m 11.5

2.10m 10.5

Table 2: Average number of bounces for each trial

Sample Calculation 1

Average bounces = ¿¿counter 1+number ¿counter 2 ¿2= 8+10

2=9 bounces

Uncertainty unchanged because all the maximum number of bounces subtracting the minimum number of bounces and dividing by 2 is smaller or equal to the given uncertainty 2 bounces.

Released Height (m) ±0.05m Average bounces ¿)

0.50m 8.8

0.90m 12.2

1.30m 15.3

1.70m 12.1

2.10m 11.1

Table 3: Average number of bounces for each variation

Sample Calculation 2

Average bounces = Trial1+Trial2+Trial3+Trial 4+Trial55

= 9+9.5+8+8.5+95

=8.8

bounces

0 0.5 1 1.5 2 2.50

2

4

6

8

10

12

14

16

18

Released Height vs Average bounces

Released Height (m)

Aver

age

boun

ces (

boun

ces)

Graph 1: Average bounces vs. Released height

From the graph, it shows a weak correlation between the best fit line and the data points. The best fit line underestimates the bounces between 0.5m to 1.3m while overestimates the bounces after 1.3 m till 2.1m. Therefore, the best way to illustrate the data seems to be separating the data points into two graphs.

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

2

4

6

8

10

12

14

16

18

20

f(x) = 11.6666666666667 x + 1.55

f(x) = 8.125 x + 4.7875

f(x) = NaN x + NaNReleased Height vs Average bounces (1)

Released Height

Aver

age

boun

ces (

boun

ces)

Graph 2: Average bounces vs. Released Height from 0.5m to 1.3m

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.20

2

4

6

8

10

12

14

16

18

20

f(x) = − 0.222222222222223 x + 13.5777777777778

f(x) = − 11.7142857142857 x + 33.1142857142857

f(x) = − 5.25 x + 21.7583333333333

Released Height vs. Average bounces (2)

Released Height

Aver

age

boun

ces (

boun

ces)

Graph 3: Average bounces vs. Released Height from 1.3m to 2.1m Graph and equations obtained by Excel 2010.

Sample Calculation 3 Line of best fit:Slope= 8.125m m/bounce

Sample Calculation 4Minimum gradient lineSelected data points (0.55, 10.8)(1.25, 13.3)

Slope = 13.3−10.81.25−0.55

=3.5714 m/bounce

Mmaximum gradient lineSelected data points (0.45,6.8)(1.35,17.3)

Slope = 17.3−6.81.35−0.45

=11.667 m/bounce

Sample Calculation 5

Uncertainty for gradient = 12|slope highest−slope lowest|=1

2|11.667−3.5714|=4.0478

m/bounce

Graph 2: Gradient = 8.125±4.0478m/bounceGradient =8.13±4.05m/bounce

Percentage uncertainty= 4.058.13

=49.8%

The percentage uncertainty is 49.8%

Graph 3: Gradient = -5.25±5.7459m/bounceGradient =-5.25±5.75 m/bounceThe percentage uncertainty is 110%

ConclusionI decided to break down the one set of data points into making two graphs in the

end. Afterwards, each graph separately demonstrated strong correlation between the best fit line and the data points, creating an excellent example of a linear model.

Graph 2 shows that for the height between 0.5 to 1.3 meters, every increase in meter will lead to an increase of 8.125 bounces of the ball. Oppositely, for the released height between 1.3 to 2.1 meters, every increase of 1 meter will lead to a decrease of 5.25 bounces f the ball.

The result of the experiment partially testifies my hypothesis which says the increase of height will lead to an increase in bounces of the ball. However, the experiment shows that after reaching around 1.3 meters, instead of increasing the bounces, the number of bounces has actually decreased. The reason this has occurred I have thought of two reasons. The first reason is because the higher the released point, the air resistance would be bigger, and because the table tennis ball is very light, this resistance must have affected the ball a lot; another reason is that the table tennis ball may be damaged by releasing from a greater height, because I did hear a different sound when the height is increased as the table tennis ball is first dropped. The ball instead of using the energy to bounce back up, it might have been instead released into sound energy and damaging the ball.

The uncertainty calculated has also been very huge, as big as 49.8% for graph 2 and 110% for graph 3. The error will be discussed in the Evaluation section. The range of values can be disperse more and definitely more variations are needed. Because each graph only has 3 datapoints, the uncertainty is huge and the reliability is also questionable.

EvaluationThe chosen type of ball had been most likely affected by air resistance as the height increases, accounts for a type of systematic error. There was obvious random error for the counters when they count the bounces, affecting the accuracy of the data and contributing to the uncertainty. The method could have been better at decreasing the uncertainty for the released height, another random error from performance.

ImprovementsNext time I will choose a different type of ball such as a tennis ball to do the

experiment. The greater mass of a tennis ball would decrease the effect of air resistance as well as the material is much durable than a table tennis ball which is made of thin plastic. Also, I would consider using a ExplorerX instead of two counters to record the number of bounces, because it would be much more accurate. Last but not least, I would come up with a better method at releasing the ball. For example, I would add a horizontal stick attached to the vertical meter tape from the wall to increase in accuracy for the release height, which would significantly decrease the uncertainty.

I think more variations should be performed for this experiment as well, after dividing the graph only 3 data points where in each illustration, decreasing the reliability of the data.