Plumb Line Mechanics Experiment

Any measurements in which the ball bounced at an angle of more than 10 degrees were ignored, as were any in which the ball hit the wall or when the person dropping the ball felt that they had accidentally given it some spin. In both parts results were taken for every 10cm between 1 and 2 metres, and 5 results were taken for each height. The heights that the ball bounced to were recorded by the height that the top of the ball reached, as this was easier to measure than the height that the bottom of the ball reached, so the results have been adjusted by 4.38cm, the diameter of the ball.

Initial Height (cm)

Height after first bounce (cm)

Average (cm)

2

3

4

5

200

73

75

75

74

73

74.0

90

64

66

65

64

64

64.6

80

55

59

57

58

58

57.4

70

46

49

49

50

49

48.6

60

42

41

42

42

42

41.8

50

34

34

33

33

33

33.4

40

25

26

27

25

26

26.8

30

17

18

18

18

17

17.6

20

09

09

09

12

10

09.8

10

00

01

99

00

01

00.2

00

91

92

92

92

91

91.6

Initial Height (cm)

Time taken for 3 bounces (s)

Average (s)

2

3

4

5

200

2.69

2.75

2.75

2.72

2.75

2.732

90

2.61

2.69

2.65

2.68

2.75

2.676

80

2.57

2.66

2.57

2.68

2.71

2.638

70

2.59

2.69

2.60

2.41

2.47

2.552

60

2.43

2.34

2.50

2.40

2.34

2.402

50

2.37

2.47

2.40

2.34

2.32

2.380

40

2.22

2.31

2.34

2.28

2.25

2.280

30

2.22

2.41

2.19

2.15

2.15

2.224

20

2.19

2.06

2.13

2.09

2.22

2.138

10

2.04

2.07

2.10

2.06

2.03

2.060

00

2.10

.90

2.00

.91

.94

.970

Presenting the Model

Part 1: Height of the First Bounce

v=speed when the ball hits the ground (ms-1)

u=speed when ball is released (ms-1)

a=acceleration due to gravity (ms-2)

s=initial distance from the ground (m)

v2=u2+2as

v2=2as

v=V(2as)

V=speed when ball has bounced to it's highest point (ms-1)

U=speed when ball leaves the ground (ms-1)

A=acceleration due to gravity(ms-2)(negative as gravity is now slowing the ball down)

S=highest point the ball bounces to (m)

e=the coefficient of restitution

V2=U2+2AS

U=eV(2as)

0=2ase2+2AS

A=-a

0=2ase2-2aS

2aS=2ase2

S=se2 ? The fact that e is known to be a constant means that S/s must

e2=S/s also be a constant. It can therefore be written as S=ks showing

e=V(S/s) that the graph of S against s must be linear.

Part 2: Time taken for 3 bounces

T=time taken for the three bounces (s)

T=t0+2t1+2t2

tn=time for ball to go from top of a bounce to the ground (s)

u=speed at the top of a bounce (always 0)

sn=top of a bounce (m)

a=acceleration due to gravity (ms-2)

sn=utn+1/2atn2

sn=1/2gtn2

2sn/g=tn2

tn=V(2sn/g)

T=t0+2t1+2t2

T=V(2s0/g)+2V(2s1/g)+2V(2s2/g)

? S=se2

sn+1=sne2

sn=s0e2n

T=V(2s0/g)+2V(2s0e2/g)+2V(2s0e4/g)

T=(1+2e+2e2)V(2s0/g)

Since the graph of T against s0 will be of the form y=kVx it will have the following shape:

In other words it will be a slightly curve towards the x-axis.

Task 1:Find e

Using the dashed lines on the graph, I wish to find the gradient of the line. The vertical line is 10.09cm long and the horizontal line is 8.12cm long. Since the two axes have the same units and scale the figures don't need to be adjusted. The gradient of the line, and hence the value of S/s, is therefore 8.12/10.09 or 0.805.

Since e=V(S/s) and S/s=0.805, we have now shown that e=V0.805 or e=0.897.

Task 2:Find the time taken up to the third bounce

Predicted Results Experimental Results

Discussion of variation in the experimental results

The fact that the second part of the experiment was so much less precise than the first leads me to believe that the majority of the variation was due to the human error in the use of the stopwatch. I believe it was so large because :

. Stopwatches in experiments are usually used to measure times of around 20s so that the effect of their inaccuracy is less important. The experiment is usually adjusted so that such measurements can be possible, but that was not really possible with our experiment as the ball would probably have come to rest within 10s.

2. The stopwatches that we used had a tendency to take 1 button press as three quick simultaneous ones, which could have resulted in about a tenth of a seconds difference either way (as the two unwanted presses add a bit of time on at the end or take a bit off at the beginning.

Most of the rest of the variation was assumably due to accidental differences in the way the ball was dropped, as giving the ball some amount of energy could not be avoided while the ball was being dropped by hand. This energy will partly have been in the form of spin, causing the ball to bounce at a bit of an angle so that the height reached was lower, and partly translational kinetic energy which will have affected how fast and hence how far it went.

The graphs for both tasks have error bars to show the spread of the results.

Comparison between the experimental results and the predictions of the model

For task 2 I have drawn in the predicted results using the value of e found in task 1. They are pleasingly similar to the experimental results and it would at first seem quite possible that that difference was merely due to some constant error on the part of the person timing, for example their having a tendency not to click the stopwatch until a second after the ball had been released. However, the fact that the difference between the two lines is small at first and gradually becomes greater suggests that some other factor is involved. As the ball is dropped from greater heights, the greater speeds it reached will have caused air resistance to have a greater effect, although I would not have thought the difference was large on a small object at such small speeds. In fact, it is possible that the unknown factor was at work in task 1, giving us a faulty value of e. Another possible (though not likely) cause is that gravity may not be 9.8ms-2 at the altitude at which we did the experiment, and this will not have had allowance made for it by our value of e, as the way it was worked out made the result independent of gravity. A higher value of g would explain why it took less time, as the ball would have bounced as high but travelled faster.

However, this slight difference does not seem to have any importance to the overall results. The two graphs have the general shapes that the model predicted and so are explained by the model.

Revision of the process

I believe that the greatest contributor to the imprecision of the experiment was the error in timing, so if it were feasible(i.e. cheap) I would use some kind of electronic equipment that detected when the ball bounced or was released. I would also have thought it better so use some simple mechanical device to drop the ball to avoid giving the ball any energy, and it may have been better to test the value of acceleration due to gravity rather than just giving it an assumed value. After all, the position of the moon has enough of a gravitational effect to cause the changing of the tides, so it may have had some noticeable effect on task 2 of the experiment, when gravity was significant.

I think testing the value of g beforehand might have helped to reduce the difference between the experimental and predicted results for task 2. I am certain that better timing equipment would reduce the size of the error bars in task 2, and I think that using some mechanical device to release the ball would slightly reduce the error bars in both tasks.