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Bouncing balls.

Extracts from this document...

Introduction

Philip Ng                                                                                                        24/03/2003

Bouncing

(Coefficient of restitution)

Introduction

Suppose two particles of masses image09.png and image10.png are moving in a straight line with speeds image16.pngand image25.png before impact and speeds image12.png and image11.png after impact, in the directions shown in figure 1. Newton’s law of impact gives

image50.png

image51.png

Where image16.png-image25.png is the relative speed with which image09.pngapproaches image10.pngand image11.png-image12.png is the relative speed with which image10.pngdraws away fromimage09.png. There is often confusion over the signs in the equation, and it is convenient to restate it in the formimage13.png

Strategy:

The initial phase of the activity involves making careful, qualitative observations of a single bounce.

Next, our team takes a tennis ball and measure quantitatively how high it bounces when dropped from a given height.    

Finally, one should consider the same ball bouncing several times and study the progressive decrease in the heights to which it goes.

image00.png

Data Collection

In order to get the coefficient of restitution of the floor, I will look for the initial height and the height after first bouncing.

In this experiment, I have made some assumption.

Assumption:  1. No Air Resistance

                            2. No friction

                            3. No external force supplied to the tennis ball

                            4. Initial velocity = 0 m/s

                            5. Ignore the spinning

                            6. Acceleration: g= a= 9.8m/s-2

                            7. The ball makes very brief contact with the table, seeming to leave I                               almost instantaneously

                            8.

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Middle

0.89

0.902

0.91

0.89

1.4

0.79

0.77

0.78

0.78

0.78

0.78

0.79

0.77

1.2

0.68

0.7

0.68

0.67

0.69

0.684

0.7

0.67

1

0.57

0.57

0.565

0.56

0.57

0.572

0.57

0.56

(The table shows the coefficient of restitution of different drop height)

image02.png

image03.png

image22.png (The graph shows the coefficient of restitution of bouncing the tennis ball on the floor)

To find the coefficient of restitution, as I have predictedimage23.png, so I useimage24.pngas my prediction whereimage23.png,image26.png. So now I vary the value of m to adjust the line to be best fit of my max, mean, and min bounce height respectively.

By using the spreadsheet,

The gradient of the lines are: Max= 0.57

                                               Min=0.55

                                               Mean= 0.56

We now have the two extremes, and the error bound is 0.56+0.1

So the coefficient of restitution is Max= 0.755

                                                      Min=0.742

                                                      Mean=0.748

Comparison

To find out the coefficient of restitution, it can be calculated by the following experiment.

Now, I am going to conduct the similar experiment as before. Drop the ball from a given height, and consider the time taken for the tennis ball to stop bouncing.

From my previous work page 2), it shows thatimage27.png,

So the second bouncing height should be image28.png

The nth bouncing height should be image29.png

image08.pngimage07.pngimage06.pngimage07.pngimage04.pngimage05.png

The time taken from drop height to the ground:

image30.png

                                     To solve:  image31.png

As I’ve got image32.pngand image17.png where v is the velocity just before first bouncing

So let u be the velocity just after first bouncing, image33.png

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Conclusion

Revision of progress

To improve the match between model and experiment, I will improve the quality of experiment. In my first experiment, to record the maximum height of the tennis ball after first bouncing, I will use the electronic equipment, like detector, instead of observation by eyes. However, I will use a heavier ball instead of tennis ball, as heavier ball has larger force acting by gravity, so it has larger potential energy than the tennis ball in the same height, so the proportion of energy loss is reduced because the total energy is larger. It can reduce the experimental error.

In my second experiment, I will again use the sensor to record the time instead of observation. It is more sensible and accurate.

The effect of these amendments is to record the more accurate data, the actual bounce height and time, so that I can get a more accurate result by my equations. However, it is difficult to do these amendments because it is complicated to set up and I do not have to do that, because my result is already very good enough. So there will not be a significant difference whether I use the electronic equipment or do it by myself.

...read more.

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