Measuring the Coefficient of Restitution of a Table Tennis Ball

Ed Moss

February 19, 2012

Measuring the Coefficient of Restitution of a Table Tennis Ball Data Collection and Processing Method A standard table tennis ball was dropped from a range on heights onto a tabletop. The ball was dropped from 9 different heights and the drop was repeated 5 times at each height. The height the ball was dropped from (H1) and the height the ball reached at the top of the first bounce (H2) were recorded and tabulated as shown below.

Raw Data H1 (cm) 21.0 27.6 33.0 41.4 46.8 58.0 70.0 86.0 100. 0

H2 H2 H2 H2 H2 H2 Repeat 1 Repeat 2 Repeat 3 Repeat 4 Repeat 5 Average (cm) (cm) (cm) (cm) (cm) (cm) 16.0 15.5 17.0 15.5 15.0 15.8 20.5 21.5 22.5 21.0 22.0 21.5 24.5 26.5 27.0 24.0 26.0 25.6 30.5 30.0 31.5 32.0 30.5 30.9 35.0 34.5 32.5 34.0 34.5 34.1 41.0 42.0 41.5 38.0 41.0 40.7 49.0 47.0 48.0 46.0 49.5 47.9 57.0 55.0 58.5 59.0 56.0 57.1 62.0

64.0

65.5

63.0

67.0

64.3

Processed Data H1 (cm) 21 28 33 41 47 58 70 86 100

H1 Uncertainty (cm) 1 1 1 1 1 1 1 1 1

H2 (cm)

H2 Uncertainty (cm)

15 22 26 32 37 41 48 57 64

3 3 3 3 3 3 3 3 3 Page 1 of 8

Ed Moss

February 19, 2012

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Ed Moss

February 19, 2012

Measuring the Coefficient of Restitution of a Table Tennis Ball 80.0

70.0 f(x) = 0.69x - 0.69 f(x) = 0.6x + 5.05 60.0

f(x) = 0.53x + 8

50.0

H2 - Height After First Bounce (cm)

40.0

30.0

20.0

10.0

0.0 0.0

20.0

40.0

60.0

80.0

100.0

120.0

H1 - Height Dropped (cm) Page 3 of 8

Ed Moss

February 19, 2012

A note on uncertainties The uncertainties in H1 (the initial drop height) are simply an estimate based on the fact that we judged the height of the ball by eye against a ruler. It would therefore be unreasonable to say we were accurate to the nearest millimeter but it is reasonable to say that the drop height we measured was within a centimeter of the actual height. The uncertainty in H2 (the height after the first bounce) is estimated by looking at the difference between the average and the individual readings (see raw data). The greatest difference between any individual reading and the average is 2.7cm (see the highlighted sections of the ‘raw data’). This shows us that a reasonable uncertainty for H2 is ±3cm.

Calculation Theory predicts that: e=

V2 V1

Where:

e = the coefficient of restitution V2 = the speed just after the bounce V1 = the speed just before the bounce

We have not measured the velocity of the ping pong ball, but from our experiment me did measure: H1 = the height the ball was dropped from H2 = the height of the ball at the top of the 1st bounce If we assume that all the gravitational potential energy of the ball is converted to kinetic energy when it is dropped then we can form the equation: 1 2 mg h1= m v 1 2 v 12=2 g h1 Similarly if we assume that, once the ball hits the tabletop, all of the kinetic energy of the ball is converted back into gravitational potential energy, then we can form another equation: 1 mg h2= m v 22 2 2

v 2 =2 g h2 Therefore we can say that:

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Ed Moss

February 19, 2012

2

v 1 2 g h1 = 2 v 2 2 g h2 v 12 h1 = v 22 h2 h2 v 22 = h1 v 12 h2 2 =e h1 2

h2=e h1 This final result is important as it allows us to use our graph to calculate a value for ‘e’. Since this equation is of the form: y=mx +c We can see that the gradient of our graph corresponds to ‘e2’. The line of best fit gives a value of ‘e2’ to be 0.6044. The maximum and minimum lines of best fit show the uncertainty in the value of ‘e2’: 2

0.53 ≤ e ≤0.6869 2

e =0.6044 ± 0.0825

( 0.6869−0.6044=0.0825 )

From this we can calculate that the percentage uncertainty in ‘e2’ is: 0.0825 × 100=13.650 0.6044 A 13.650% uncertainty in ‘e2’means there with be a 6.825% uncertainty in ‘e’. Therefore we can say that. e=0.7774 ± 0.05305

Now correcting this t a reasonable number of significant figures: e=0. 78 ±0.05

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Ed Moss

February 19, 2012

Conclusion and Evaluation Conclusion When the data is displayed on the graph it clearly shows that there is a linear relationship between H1 and H2. There is a straight line of best fit which passes through all of the points (allowing for errors). Our graph implies that H1 is proportional to H2 but it does not imply direct proportionality as the line of best fit does not pass particularly close to the origin. The value of ‘e’ we calculated was ‘0.78±0.05’, however this is from the literature value, the below paragraph is from a Wikipedia article: ‘The international rules specify that the game is played with a light 2.7 gram, 40mm diameter ball. The rules say that the ball shall bounce up 24– 26cm when dropped from a height of 30.5cm on to a standard steel block thereby having a coefficient of restitution of 0.89 to 0.92.’1 If we take the true coefficient to restitution to be 0.905 then our error is 16.4%. This is much greater than our estimated error of 6.8%. The reasons behind this discrepancy are explored in the evaluation.

Evaluation It is first important to comment on the quality of the data, although our final result was wrong the data itself seems not to have any anomalous points. At each height there was a small data spread (always less than ±3cm) but overall the data seems coherent. The uncertainty comes from the fact that we are only judging the height of the ball by eye, and so, depending on where you look at the ball from, it may seem to be at different heights. It is also very difficult to measure H2 as the ball is moving relatively quickly and so it is difficult to determine where precisely the top of the bounce is. There is undoubtedly a random error in both H1 and H2 but this alone does not seem to be able to explain why our result is so far off, indeed in our calculations we factored in uncertainties in H1 and H2 and the result was still wrong. The discrepancy between our result and the actual result is probably due to a systematic error. Below are three potential errors in the experiment which may have caused the discrepancy. The table tennis ball we were using may have been old or not official. The value of 0.89-0.92 describes the resistivity of competition standard ping pong balls. The ball we were using may not have been made to completion standard, i.e. it may not be as bouncy as a completion ball. If the ball was indeed less bouncy than a completion ball then (as we recorded) its coefficient of resistivity would be less than 0.89. 1“Table Tennis.” Wikipedia Online. http://en.wikipedia.org/wiki/Table_tennis (accessed 14/02/12) Page 6 of 8

Ed Moss

February 19, 2012

The surface we were dropping the ball onto may have damped the bounce of the ball. The official measurement is made using a steel plate, but we performed our experiment on a laminate work surface. If the worktop is a better damper than the steel the ball would not bounce as high and thus ‘e’ would be less than the official value. The third (and in my opinion most likely) reason why our value of ‘e’ was so far off was that some of the assumptions made in our calculation were incorrect. We assumed that when the ball was dropped all its potential energy was converted to kinetic energy. However, when the ball is dropped from some of the higher heights, it is travelling at a reasonably high speed when it falls. This means that there is a significant air resistance acting upon the ball, i.e. some of the potential energy is lost to heat as the ball falls. This means that the ball is not travelling as fast as it hits the surface and so does not bounce as high. I.e. H2 is lower than it should be. For example, at a height of 100cm, we recorded H2 to be 64 whereas if air resistance was not a factor it would probably be closer to 80 or 90. This means that our line of best fit is not as steep as it should be, so the gradient is smaller, so the calculated value of ‘e’ is smaller. The impact of air resistance is probably the main reason why the data does not give us a correct value of ‘e’. It is a systematic error as all the balls will bounce less high as they would have done if there was no air resistance. However this systematic error does not shift the whole line of best fit as it affects the balls dropped from higher points than those dropped from lower points2. It simply distorts the data of the higher balls and so means the gradient is less than it should be. Below is a graph of the experiment if we only use the first 3 lowest readings. At these heights the effect of air resistance is negligible. Using this line of best fit the calculated value of ‘e’ is 0.905, which agrees with the literature value of 0.89-0.92.

2 This is due to the fact that the amount of air resistance depends on the velocity squared. Page 7 of 8

Ed Moss

February 19, 2012

Measuring the Coefficient of Restitution of a Table Tennis Ball 30.0 25.0

f(x) = 0.82x - 1.29

20.0 H2 - Height After First Bounce (cm)

15.0 10.0 5.0 0.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 H1 - Height Dropped (cm)

Improving the investigation In order to reduce the random error in H2 I would suggest recording the bounce of the ball with a high speed camera. The footage could then be replayed in slow motion and it would be far easier to see exactly what height the ball reached. In order to reduce the error in reading off the height of the ball against the ruler, one could use a motion sensor positioned above the ball. If you know the distance from the motion sensor the surface then you can use the data from the motion sensor to calculate the height of the ball more precisely. A motion sensor connected to a computer also is a good idea as if you plotted a graph of the balls height it will be clear to see where the top of the bounce was. To remove the systematic errors in the experiment, I would suggest using a band new official table tennis ball and bouncing the balls off a steel plate as is the official procedure. I would also suggest using a motion sensor connected to a computer to record the speed of the ball before and after the bounce, then I would plot V2 against V1 and the gradient of that line of best fit would give a value for ‘e’. This is a better procedure as in this example we do not have to make any assumption about the potential and kinetic energy of the ball.

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