Extended Experimental Investigation

Extended Experimental Investigation

The Restitutions of a Bouncing Ball

Abstract: This extended experimental investigation is aimed at looking closely into the physics of a bouncing ball. It will look at the different factors that make a ball bounce higher like the size, material or internal pressure. With the hypothesis that the ball that have better elastic properties will bounce better and the balls with the higher internal pressure would also bounce better. The experiment was a ball being dropped from the height of 1 metre and recorded by a video camera for later analysis. Different variable were changed to get a fair amount of data to analyse and discuss in the report. Furthermore the results proved the hypothesis correct and opened discussion to possible modifications to the experiment. In addition the best performing ball was analysed and conclusions were made on why it bounced so well. Finally, possible relevant scenarios and their possible outcome were envisioned in the Discussion.

GRIESHABER, Michael

4/5/2014

Introduction:The bouncing of a ball is something everybody has done in some point in their childhood. The motion of a bouncing ball is not something you really think about that much. You throw the ball down and it bounces back up and each time it bounces it doesnt bounce as high as the bounce before. The motion of a bouncing ball is an interesting topic to investigate, indicating numerous interesting dynamics principles correlated with acceleration, momentum, and energy.To begin this explanation let's first consider what happens to a typical high bounce ball that is dropped under the influence of gravity. The motion of a bouncing ball can be broken up into seven distinct stages, where the motion of the ball is analysed before, during, and after impact. For simplicity assume that the surface is hard and rigid, and ignore air resistances effect on the balls motion through the air. The geometric centre of the ball is defined as point C, the velocity of point C as V, and the acceleration of point C as A. Assume that the ball has uniform density; this means that point C is also the centre of mass. Stages of a bouncing ball falling vertically under influence of gravity:Stage 1 In the first stage, the ball has gravitational potential energy. The formula for gravitational potential energy is:. With m being the mass of the ball and g being the acceleration due to gravity and h the height at which the ball is dropped from. When the ball is let go it falls vertically downward under the influence of gravity (g). And the gravitational potential energy is transformed into kinetic energy as it falls. The amount of gravitational potential energy and kinetic energy at different stages during the drop is proportional to the height at which the ball is e.g. if the ball is half way between the height it was dropped and the surface it will bounce off, the ball has 50% potential energy and 50% kinetic energy. The velocity V points downward. The acceleration A also points downward. The magnitude of A is equal to g, if the air resistance is ignored. Acceleration due to gravity is approximately 9.8 m/s2. (Note all masses must be recorded in kilograms and all heights must be in metres)

Stage 2

In the second stage, the ball starts to make contact with the surface and all the gravitational potential energy has now being transformed into kinetic energy. So the equation: (m represents mass and v represents velocity) is used to work out the kinetic energy. But we already know that 100% of the potential energy has being transformed into kinetic energy. So we know the kinetic energy but not the velocity. The equation can simply be rearranged to work out the velocity at this stage: . The ball carries on falling vertically downward under the influence of gravity. The velocity V and acceleration a (equal to g) both continue to point downward.

Stage 3

In this stage, the ball has slowed down. The velocity V is still pointing downward. However, the ball has deformed sufficiently such that the acceleration a is now pointing upward. This means that the ball has deformed enough such that it's pushing against the surface with a force greater than its own weight. As a result, the acceleration a is pointing upward. The kinetic energy begins to transform into elastic potential energy as the ball deforms. The deformation is like a spring being compressed except the ball deforms in 3 dimensions instead of 1 as a spring. The amount a ball deforms depends on the type of material a ball is made up of, the shape and the air pressure inside the ball (if it is hollow). This is where the energy is lost as sound or heat e.g. when you hear a ball bounce and when a ball becomes warm after using it for a long period of time.

Stage 4

In this stage, the ball has reached its maximum deformation. As a result, the acceleration a is still pointing upward, and the velocity V is zero. This means that point C is at its lowest point. The ball now has transformed all the kinetic energy into elastic potential energy and obviously has no gravitational potential energy at this point.

Stage 5

In this stage, the ball velocity V is increasing and pointing upward since the ball is now in the rebounding stage. As a result, the ball is less deformed than in the previous stage, but is still deformed enough such that it's pushing against the surface with a force greater than its own weight. This means that the acceleration a is still pointing upward. The elastic potential energy is now transforming back into kinetic energy because the restoring force will return the ball to its original shape. (Note there will not be as much kinetic energy as there was when it came into contact with the surface because the energy was lost as heat and sound energy)

Stage 6

In this stage, the ball is barely touching the surface. The velocity V is still pointing upward since the ball is still in the rebounding stage. However, since the ball is no longer deformed it has essentially zero contact force with the surface. This means that the only force acting on the ball is gravity. As a result, the acceleration a is now pointing downward, and the upward velocity V is now decreasing. The ball now has 100% of the remaining energy as kinetic energy. Therefore it will not bounce as high as the point it was dropped from.

Stage 7 In this stage, the ball has fully rebounded and has lifted off from the surface. The velocity V is still pointing upward, and the acceleration a is still pointing downward since the only force acting on the ball in this stage is gravity. The kinetic energy is transforming into gravitational potential energy until it reaches its peak where there is 100% gravitational potential energy and the whole process happens again and continues to bounce until there is not enough elastic potential energy to lift the weight of the ball off the ground. Due to the fact that gravity (9.8m/s2) is greater than the acceleration the elastic potential energy produces.

Aim: The aim of this extended experimental investigation is to investigate and discuss the effects the size, material and internal pressure has on the height of the restitutions, the coefficient of restitution and change in velocity before and after impact when the controlled variables are kept the same. Research questions: How does the size affect the height of the bounce? The size of the ball does not give the ball any advantages in it capability to bounce higher. The only affect it has is the air resistance of having a larger surface area. How does the material affect the height of the bounce? The material of a ball determines how high it will bounce. A ball that has material with high elastic properties is going to bounce higher that a ball made of a more ridged material with low elastic properties. How does the internal pressure affect the height of the bounce? The internal pressure of a ball (that is pumpable) should have an effect on the height of restitution of the ball. If the ball is pumped to it recommended internal pressure it should have high elastic properties depending what the purpose of the ball is. When the balls internal pressure is below its recommended internal pressure it should not bounce as high as it did when it had the recommended internal pressure. The independent variable was the different balls that were dropped. The dependent variables were the height of the bounces and the coefficient of restitution and the velocity before and after impact. The controlled variables were the height of the drop which was kept at 1 metre; the surface the ball was dropped on which was wood covered in lino and the spot where the ball was dropped was kept the same throughout the experiments. The uncontrolled variables were the way the ball was released from the hand i.e. sometimes the ball was released with spin, the breeze and the temperature.Independent Variable

The ball that was bounced

Dependent Variables

The height at which it bounced to and the coefficient of restitution and the velocity before and after impact

Controlled Variables

Drop Height (1 metre)

Surface the ball was dropped on

The ball was dropped on the exact same place every time

Uncontrolled Variables

The way the ball was released from the hand sometimes it had spin

The breeze that came through the door

The temperature

Hypothesis: For the small all the small balls it was hypothesized that as the material of the ball was less ridged and more elastic the height of the balls restitution would increase. For the large ball it was hypothesized that as the internal pressure was reduced from the recommended pressure to 50% of the internal pressure the height of the restitution would decrease.Materials: Small BallsLarge BallsOther

Ping Pong ballBasketballCamera

High Bounce ballSoccer ballTripod

Large High Bounce ballNetball1 meter ruler

Small Tennis ballSoft VolleyballWooden backboard

Large Tennis ballHard VolleyballLino covered wooden floor

Sponge BaseballPressure gauge

Rubber Cricket ballElectric ball pump

Blu-tac

Tracker (Visual analysing program)

Procedure:1. All equipment for experiments was collect.2. The 1 metre ruler was connected to the back board with blu-tac so it was straight and one end of it was at the bottom of the backboard.3. The backboard with the ruler attached to it was place vertically against a wall.4. All the balls were weighed and their masses were recorded.5. The camera was set on a tripod and was setup so that it could capture the motion of the bouncing ball.6. The camera was tested if it worked properly and the SD card was removed from the camera and placed into a laptop to ensure that it recorded and was able to transfer the file.7. A random ball was selected and dropped from 1 metre and recorded then analysed on tracker as a test.8. The first of the small balls the ping pong ball was dropped from 1 metre onto the lino covered wooden floor. The bounces were recorded by the camera. This was completed 3 times to rule out any experimental error.9. The 7th step was repeated for the other 6 small balls.10. The large balls were pumped to a little more than their recommended psi with an electric ball pump and then the excess was released with pressure gauge until they precisely their recommended psi.11. The first of the large balls, the soccer ball was dropped from 1 metre onto the lino covered wooden floor. The bounces were recorded by the camera. This was done 3 times to rule out experimental error.12. The Soccer Ball was deflated to 50% of its recommended psi and then step 11 was repeated.13. Steps 11 and 12 were repeated for the other 4 large balls.14. All the data was transferred from the SD card 15. All the files were analysed on tracker

Results: Small Balls:How to work out Coefficient of restitution:

Ping pong ball 1st bounce:

Ping pong ball 2nd bounce:

Note: these steps were repeated for all small balls but for simplicity only the working out for the ping pong ball was shown.

Height of Restitution and Coefficient of Restitution

Ping Pong ballHigh Bounce ballLarge High Bounce ballSmall Tennis ballLarge Tennis ballSponge BaseballRubber Cricket ball

Drop height1.001.001.001.001.001.001.00

Bounce 1 height 0.570.740.750.580.510.520.47

Bounce 2 Height0.340.560.570.360.260.290.20

Coefficient of restitution 10.750.860.870.760.710.720.69

Coefficient of restitution 20.770.870.870.790.710.710.65

How to work out the Gravitational potential energy:

Gravitational potential energy of ping pong ball at drop height:

How to work out the Kinetic energy at stage 2: But the Kinetic energy at stage 2 is already known because it is the same as the gravitational potential energy. So the only other unknown in the equation is v, velocity. Therefore the equation is rearranged like so: Velocity of ping pong ball at stage 2:

Gravitational potential energy of ping pong ball at peak of first bounce:

Work out the velocity the same way as for the initial drop:

To work out the loss of energy take the of the first bounce away from the at the drop height. Work out the amount of energy lost as heat and sound:

Velocity Before and After Impact, Mass Gravitational Potential Energy,

Ping Pong ballHigh Bounce ballLarge High Bounce ballSmall Tennis ballLarge Tennis ballSponge BaseballRubber Cricket ball

Velocity Before impact (m/s)4.434.434.434.434.434.434.43

Velocity After impact (m/s)3.343.813.833.373.093.163.04

Mass (kilograms)0.00260.04560.12850.05540.20640.06060.1935

(Joules)0.0255 0.4469 1.2593 0.5429 2.0227 0.5939 1.8963

at stage 2(Joules)0.0255 0.4469 1.2593 0.5429 2.0227 0.5939 1.8963

Bounce 1 (Joules)0.0145 0.3307 0.9445 0.3149 1.0316 0.3029 0.8913

at stage 6 (Joules)0.0145 0.3307 0.9445 0.3149 1.0316 0.3029 0.8913

Energy lost as heat and sound (Joules)0.2403 0.1162 0.3148 0.2280 0.9911 0.2910 1.0050

Drop time (seconds)0.450.450.450.450.450.450.45

Large Balls:Soccer ballNetballBasketballHard VolleyballSoft Volleyball

Drop Height1.001.001.001.001.00

Bounce 1 height 100%0.590.670.780.630.82

Bounce 1 height 50%0.520.540.680.530.78

Bounce 2 height 100%0.360.450.590.400.69

Bounce 2 height 50%0.270.310.480.290.62

Coefficient of restitution 1 100%0.770.820.880.790.91

Coefficient of restitution 1 50%0.720.730.820.730.88

Coefficient of restitution 2 100%0.780.820.870.800.92

Coefficient of restitution 2 50%0.720.760.840.740.89

Velocity Before impact (m/s) 100%4.434.434.434.434.43

Velocity Before impact (m/s) 50%4.434.434.434.434.43

Velocity After impact (m/s) 100%3.403.623.913.514.01

Velocity After impact (m/s) 50%3.193.253.653.223.91

Mass (kilograms) 100 % and 50% 0.42650.42200.58920.26410.2196

(Joules) 100% and 50%4.17974.13565.77422.58822.1521

at stage 2(Joules) 100% and 50%4.17974.13565.77422.58822.1521

Bounce 1 (Joules) 100%2.46602.77094.50381.63061.7647

Bounce 1 (Joules) 50%2.17342.23323.92641.37171.6786

at stage 6 (Joules) 100%2.46602.77094.50381.63061.7647

at stage 6 (Joules) 50%2.17342.23323.92641.37171.6786

Energy lost as heat and sound (Joules)100%1.71371.36471.27040.95760.3874

Energy lost as heat and sound (Joules) 50%2.00621.90241.84781.21650.4735

Drop time (seconds)0.450.450.450.450.45

Pressure 100% (psi)10101053.8

Pressure 50% (psi)5552.51.9

Note: the methods for working out the velocity,, energy loss and coefficient of restitution were the same as in to small balls therefore it was unnecessary to repeat the example of the working out for the large balls.Note: The pressure was measured with a pressure gauge.Note: Rows shaded the same colour are supposed to have the same or very similar values.

Displacement time graph:Comparing two small balls:Small Tennis Ball Time vs. Height

Height (metres)

Time (seconds)

Rubber Cricket Ball Time vs. Height

Height (metres)

Time (seconds)

Comparing large balls at 100% recommended internal pressure and 50% recommended internal pressure:100% Netball Time vs. Height

Height (metres)

Time (seconds)

50% Netball Time vs. Height

Height (metres)

Time (seconds)

Discussion: The investigation was aimed at answering the research question: How does the size, material and internal pressure have an effect on the height of the restitutions, the coefficient restitution and change in velocity before and after impact when the height and surface the ball is dropped onto are kept the same. The coefficient of restitution was worked out by first having the height of the drop and the height of the first bounce. The formula is derived from the change in velocity of an object in a collision. In this application, a bouncing ball, the ratio is used. The kinetic energy or the ball is proportional to the velocity squared of the ball. Therefore the coefficient of restitution is equal to the square root of the final kinetic energy over the initial kinetic. . Since the amount of kinetic energy at stage 2 is equal to the gravitational potential energy at the top of the drop/bounce. Therefore the coefficient of restitution can we written as . And the formula for gravitational potential energy can be substituted in. The mass and acceleration due to gravity cancel out because they are the same on both sides of the division line so the formula is now: and is replaced with to make the formula identifiable as the coefficient or restitution: . This formula was used to find the coefficient of restitution of all the different ball tested and determined which ball bounced better. The hypothesis for the small balls that as the material of the ball was less ridged and more elastic the height of the balls restitution would increase. The data that was collected supported this hypothesis with the large high bounce ball bouncing the highest at 0.75 metres and the solid centred rubber cricket ball bouncing the lowest at 0.47 metres. The properties of the large high bounce such as the air filled centre and the elastic rubber shell optimised the conversion of kinetic energy into elastic energy and back into kinetic with little energy loss (0.3148 J). Whereas the solid centred rubber cricket ball had poor elastic qualities because it was solid and did not deform very much when it came into contact with the ground therefore losing a lot of energy as heat and sound (1.0050 J). The hypothesis for the large balls that as the internal pressure was reduced from the recommended pressure to 50% of the recommended pressure the height of the restitution would decrease. Once again the data supported the hypothesis with every case showing the ball with half the recommended internal pressure bounce lower than the ball at the recommended amount. This was most clearly shown in the netball where the first bounce of the 100% ball was 0.67 metres and the first bounce of the 50% ball was 0.54 metres which is 0.13 metres change in height.There were not many serious anomalies in the data that was collected. Many of the anomalies that occurred were in the coefficient of restitution part of the table. The coefficient of restitution is supposed to be exactly the same for the first bounce and the second bounce theoretically. For a few cases like the large high bounce ball, large tennis ball and the 100% recommended internal pressure netball were exactly the same and proved the theory to be correct. But the majority of the coefficients were off by 0.01. This is not a bad result because most of the data pointed towards the theory being correct. There were a few anomalous values such as the small tennis ball, rubber cricket ball and 50% recommended internal pressure netball. These values were out by 0.03. While these values are not way off they still show that there were some experimental or analytical errors. One of the experimental errors was that the ball was dropped by a human each time. A human hand shakes and is not completely still this could affect how the ball dropped. As well when releasing the ball if one half of the ball is released before the other it may put spin on the ball and cause it to bounce irregularly. Another human forced error was that the ball was not dropped from exactly 1 metre each time. The person who was dropping the ball did not always check at eye-level if the bottom of the ball lined up with the 1 metre mark. The temperature of the balls could have affected the way they bounced. Like with a squash ball how you warm up the ball before a game so it bounces better so might of some balls bounced better if they were left in the sun in the sunroom before they were tested compared to other ball that might of being under the shelf away from the warm sunlight. Another slight but possible error is the draft that comes in the door from the hallway that may have affected the way the ball bounced by increasing the air resistance therefore the ball would lose more energy and not bounce as high. To solve these problems and model a more accurate experiment next time a few thing would need to refine in the experiment. Firstly I would recommend that a type mechanical structure be made to release the ball when a string is pulled that way it eliminated the error of the human putting spin on the ball and also the issue of not dropping the ball from the exact same height each time. Another amendment would be to place all the balls that are going to be tested in the exact same spot out of the sun (unless this is one of the variables) when storing them this way they will all be relatively the same temperature when you test them. Finally to eliminate the factor of a draft affecting the testing, do the experiments away from any open doors or windows and make sure the fans or air-conditioning are turned off be for the tests commence.Ball at bottomBall at top1 metre rulerCamera

One of the analytical errors was the position of the camera. When the videos were analysed on tracker the bottom of the balls did not line up with the 1 metre mark when the balls were being dropped, nor did the bottom of the ball line up with the base of the 1 metre ruler (as shown in the above diagram). Another analytical error was the fact that the video became very blurry when it was shown in slow motion and it was hard to determine exactly where the ball was because it appeared in 3 different places in the same frame. Furthermore the camera that was used did not show enough frames per second to sometimes see the ball hit the ground. One frame the ball would be just about to impact then the next frame it would be already higher than the last frame traveling away from ground. Ball at bottomBall at top1 metre rulerCamera

This diagram shows, by increasing the distance between the camera and the ball bouncing the angle of difference between the bottom of the ball and the 1 metre mark decreases. This makes it more accurate when analysing on tracker. To solve the other problems of the image being blurred and the camera not recording enough frames per second it would be advised to buy a better camera with a higher frames per second rate and a better resolution.The ball that bounced the highest and had the highest coefficient of restitution of 0.92 was the soft volleyball when it was pumped to its assumed recommended internal pressure. The actual recommended internal pressure was not give therefore it was pumped until if felt firm and that was then labelled as its recommended internal pressure. The reason this ball bounced so high was because of the material it was made out of and how thick the material. The manufacturer had intended it to bounce high off the wrists of a volleyball player so that it would give maximum time for one of the players team mates to get under the ball and hit it over the net. The material was a 5mm thick synthetic rubber which stretched easily and had excellent elastic properties. When the ball was bounced it deformed in to an elliptical shape (as shown in insert). It was shown to lose to least amount of energy as heat and sound of the small balls because it conserved its energy the best.Insert- Soft Volleyball deforms as it reaches its maximum deformation

The difference between the 100% recommended internal pressure balls and the 50% internal pressure balls was slight to the naked eye but on tracker it was clear that the 50% balls were bouncing lower than the 100% ball. The reason for this is because the air pressure inside in greatly less; the forces pushing out on the ball, giving it its elastic properties have diminished. The lower air pressure means that the ball is less effective in transforming kinetic energy in to elastic potential energy the back into kinetic energy and lost more energy as heat and sound energy as a result. Not all of the purposes for the balls were so the ball bounced higher some balls are designed so the bounce minimally. Like the rubber cricket which performed very poorly not even bouncing to 50% of the drop height. The reason this ball is designed not to bounce much is because in cricket where the bowler bowls the ball from anything from 50km/h to 160km/h the batsman still needs to be able to hit the ball. If a ball with a higher coefficient of restitution was used the ball would bounce over the batsmans head every ball for 4 byes. Because when the ball is released from the bowlers hand there is enormous amounts of kinetic energy and it doesnt need a very elastic ball to maintain the high speed of the bowl.One possible scenario for this test being practically applied would be if a high bounce ball company was trying to get their ball to bounce higher than the other companies balls. There would be tests of all different types of balls and the properties of the balls that bounced the best would be noted then there would be discussion on how to utilise all of the things that help a ball conserve its energy as it bounced. A few possible point they night discuss would be the air pressure inside the ball, how high was too high? Another might be If one material stretches the best and another material compresses the best, why not made a dual layered ball with a stretchy outer layer and a compactable inter layer to maximise the effects of both and work out the right ratio of each layer and how thick they should be. This idea came from roman bow and arrows how they used to make the outer side of deer hind leg tendons and the inner side of a beasts tusk. After these discussions they might trial some prototypes and create the worlds bounciest ball.Conclusion: In conclusion the hypothesis and research question were confirmed and answered. The size, material and internal pressures effect on the height of the restitutions, the coefficient of restitution and the change in velocity before and after impact were investigated and discussed in this extended experimental investigation. Experimental and analytical errors were noted and recommendations were given to avoid these problems in the future. The reasons for the most efficient balls high coefficient of restitution were discussed as well as why other balls had so poor coefficient of restitutions. The difference between the 100% recommended internal pressure and the 50% ones were talked over and possible relevant scenarios and their possible outcomes were envisioned.

Bibliography: http://www.real-world-physics-problems.com/bouncing-ball-physics.html (accessed 2/4/2014)http://seniorphysics.com/physics/eei.html (accessed 2/4/2014)http://www.racquetresearch.com/coeffici.htm (accessed on 13/5/2014)http://en.wikipedia.org/wiki/Elastic_energy (accessed on 8/6/2014)http://en.wikipedia.org/wiki/Coefficient_of_restitution (accessed on 27/5/2014)How to do a deadly EEI- By Dr Richard Walding, Reseach Fellow, School of Science, Griffith University (given out by Mr Bovey)On the move A study of physics behind moving objects understanding the relationship between force energy and motion (given out by Mr Bovey)On the move- Using digital data logging and data analysis apparatus and software (given out by Mr Bovey)Physics A Contextual Approach, pages 237-240 (given out by Mr Bovey)