The mass of the ball will affect the height the ball bounces to because it affects the balls starting energy.
GPE=Mass (kg) × Gravitational Field Strength (N/Kg) × Height (m)
If we let mass = m
Gravitational Field Strength = g
Height = h1
An increase in m, assuming g and h1 stay constant, results in an increase in m × g × h1 which results in an increase in GPE.
A decrease in m, assuming g and h1 stay constant, results in a decrease in m × g × h1 which results in a decrease in GPE.
Therefore any change to the weight of the ball will affect the energy the ball has initially, which, as previously stated, affects the height to which the ball bounces.
Also the mass of the ball affects the chances of the ball reaching its terminal velocity. If the mass of the ball is heavier the weight is heavier (weight = m×g) and downward force acting upon the ball is greater as well. This means for the ball to reach terminal velocity the drag force has to be bigger and for the drag force to be bigger the ball has to fall faster (so that more air particles hit the ball every second). Therefore the heavier the ball is, the faster its terminal velocity. This means that if a heavier ball is to be used then it will need to be dropped from higher to reach its terminal velocity.
Air pressure will affect the ball’s fall slightly as the concentration of air particles per cubic meter varies with air pressure. The higher the air pressure the more air particles per cubic meter. The more particles per cubic meter, the more drag acting upon the ball.
The material ball is made from will affect the ball as if it is smooth then the drag will be significantly less than if it is rough. If the drag is less the ball will fall faster and is less likely to reach its terminal velocity. Also it will affect its bouncing properties.
The surface onto which the ball is dropped will affect the height to which the ball bounces because for any two objects that collide, the properties of both determine the percentage of the kinetic energy either possesses approaching the collision that is conserved subsequent to the collision taking place (Coefficient to restitution) discounting the effects of air resistance. For a falling object the Coefficient to restitution (CR) is equal to the velocity squared as the object is travelling at as it leaves the floor (v22) divided by the velocity squared as it hits the floor (v12):
CR= v22/ v12
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If a ball is dropped in a vacuum. The ball starts at height h1. GPE = m × h1 × g
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Energy ball starts with = mh1g
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No energy is lost when the ball is falling; there is no air resistance, so no Thermal Energy is produced. Therefore the energy that the ball hits the floor with = mh1g
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The proportion of energy lost when ball hits the floor = The Coefficient to the restitution of the two objects (CR)
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Energy ball leaves the floor with = CR (mh1g)
- All of the energy that the ball leaves the floor with is converted back into GPE
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GPE = CR (mh1g)
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At h2 the ball has energy CR (mh1g) or mh2g
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Therefore CRmh1g = mh2g
CR h1 = h2
CR = h2/ h1
If dropping a ball in a vacuum all you need to know in order to know how high the ball will bounce to is h1 and CR. CR can be found out by looking at a graph, the gradient, as a percentage of 1 gives the amount of energy conserved and therefore CR can be found without knowing v22 or v12.
This applies to a ball falling in a vacuum. When a ball drops in air there is air resistance to which the ball loses energy in the form of thermal energy. The energy that the ball hits the floor with is kinetic energy. KE = 1/2mv² where m = mass and v = velocity
Thus:
1/2mv² = mhg - thermal energy (lost as a result of drag)
As drag is a squared function, proportional to the square of the velocity, it is impossible to calculate the velocity that the ball hits the floor at.
However when the ball is dropped from a relatively low height, drag ≈ 0. This means that we can approximately calculate the amount of energy that the ball conserves as it hits the floor and therefore the height to which it will bounce for any given height in a vacuum. In air considerations have to be taken into account such as air resistance but even so the rough height to which it will bounce to can be predicted before dropping the ball. The difference between the predicted height and the actual height will provide evidence as to how air resistance affects the flight of the ball.
Fair Test:
Height will be the variable that we will vary. This is because it is the easiest and quickest variable to alter. The maximum height will have to be less than two meters as that is the maximum height that the equipment allows. I plan to collect at least ten results as this will make the conclusion and graph I am able to draw from the experiment more accurate than if I had less results than ten. This will mean that I will have to have the interval between the different heights from which the ball is dropped from less than 20cm, probably at 10cm.
The exact interval will be determined after the preliminary experiment, as will the number of heights that the ball will be dropped from. The decision for the size of interval and the amount of results collected will depend upon the time taken to conduct the experiment and any other factors that may become apparent during the preliminary experiment.
Parallax error will be avoided by dropping the ball one time that will not be measured and placing a blob of blue tack onto the meter rule at the approximate height it bounced to. Then when dropping the ball again eye level will be kept level with the blue tack thus avoiding parallax errors.
The weight and material of the ball will be kept the same throughout the experiment by using the same ball. This will be a table tennis ball. The ball weighs exactly 2.5g.
Air density will not change enough to affect the flight of the ball seeing as all the results will be collected during a brief period on one day. Temperature will not affect the balls bounce either as the experiment will be conducted at room temperature, thus not allowing the floor to get cold and in doing so alter its affect upon the ball on impact.
The surface onto which the ball is dropped upon will be kept the same. It will be vinyl tiling. The same square of tiling will be used throughout the experiment so that inconsistencies between different floor tiles do not affect results.
Safety:
- The clamp stand will be clamped down to the desk using a g-clamp to prevent it falling over and causing possible injuries.
- No balls will be allowed to roll around upon the floor creating possible tripping hazards
- Safety spectacles will be worn at all times
Preliminary Experiment:
The experiment was conducted as the method (below) states. Five repeats were done as it was deemed that an average of the middle three was reasonably accurate.
Preliminary Experiment results table
From this it can be seen that using the average of the middle three results is more accurate than using the average of all five, as it automatically discounts most anomalies. For instance result 3 from 200cm was an anomaly but was not taken into account when taking the average of the middle three. This therefore provides accurate and reliable results.
During the preliminary experiment it was established that time was not an important factor that had to be taken into account when deciding how many different heights to drop the ball from and the interval between those heights. This is because the experiment is a very short and simple one to carry out and if conducted efficiently can be completed easily within the time span allowed for collecting evidence.
It was decided that the first drop would start at 2m off the floor and then move down in intervals of 10cm to 10 cm off the floor.
Apparatus: Clamp stand, meter rule ×2, table tennis ball, desk
Method: The apparatus will be set up as shown:
The ball will then be dropped.
The ball will then bounce:
h1 = The distance between the bottom of the ball before it is dropped and the ground.
h2 = The distance between the bottom of the ball at the top of its arc after bouncing and the ground.
H is the height of the ball before it is dropped. h1 will start at 2m and then move down in intervals of 10cm to 10cm. h is the height of the balls bounce. h1 and h2 are from the bottom of the ball as it hits the floor to the bottom of the ball at the top of its arc after bouncing. This is because it is easiest as the figure read of the meter rule is the result. Also the ball flattening upon impact doesn’t have to be taken into account whereas if one was measuring from the top of the ball as it hits the floor to the top of the ball before dropping it or at the top of its arc after bouncing or the middle of the ball as it hits the ground to the middle of the ball before dropping it or the middle of the ball at the top of its bounce then the fact that the ball flattens momentarily on impact with the floor would have to be taken into account.
As the ball flattening upon impact with the floor is not visible as it happens so quickly it would be almost impossible to measure the size of the ball on impact with the floor. This is why it h2 will be from the bottom of the ball as it hits the floor to the bottom of the ball at the top of its arc after bouncing.
This will be repeated five times, possibly more (for accuracy), for each height and the top and bottom results will be discounted. An average will then be taken. This will be called the average of the middle three repeats. This will hopefully discount any anomalies automatically and leave us with three accurate and reliable results.
Results:
Conclusion:
The higher the height from which the ball was dropped from, the higher the height to which it bounced. When the ball was dropped from the higher heights the ball began to show signs of reaching its terminal velocity before it reaches the ground. These both support my prediction and show that my prediction was correct. The ball did not appear to reach its terminal velocity which also supports my prediction.
As the height from which the ball was dropped from was increased, the GPE energy that the ball possessed before being dropped also increased. The more energy that the ball possessed before being dropped, the more energy was converted into KE while the ball fell. The more KE that the ball possessed as it hit the floor, the more that was transferred into elastic potential energy and back into KE. The more KE the ball leaves the floor with the longer it takes to stop due to the force of gravity and return back to the floor again. The longer it takes to stop, the higher it bounces to.
The ball showed signs of reaching terminal velocity because the drag force began to approach the force gravity exerts upon the mass of the ball. As it did so the downward force was partially balanced out by the upwards force of drag, increasingly so the closer the ball got to its terminal velocity. The ball did not reach terminal velocity however as it did not have enough time to accelerate to its terminal velocity.
Calculating the coefficient to restitution of ball hitting the floor:
The coefficient to restitution can be found out from a graph of h1 against h2. As CR = h2/ h1 it follows the gradient of the graph change in h2 / change in h1 = CR
Below is the graph of h1 against h2. For the lowest three points air resistance is approximately equal to zero due to the ball having a low velocity, as it was dropped from a low height, and therefore hits less air particles per second than a ball traveling at a faster speed. Air resistance exists but does not affect the velocity of the ball significantly.
Gradient = 0.7
Therefore the coefficient to restitution = 0.7
This is further proved by:
Height the ball bounced to (average of middle three) when dropped from:
10cm = 6.7 6.7 = 66% of 10 (to the nearest percent)
20cm = 14.0 14.0 = 70% of 20 (to the nearest percent)
30cm = 22.3 22.3 = 74% of 30 (to the nearest percent)
Average of the three percentages = 70%
Therefore the coefficient to restitution = 0.7
30% of the energy that the ball hits the floor wit is lost. 70% is retained.
If the coefficient to restitution = 0.7, a ball dropping from h1 in a vacuum would reach the height of 0.7 h1 after bouncing.
A ball that is dropped in air however is subjected to air resistance which affects the height to which it bounces. As the ball falls it hits against air particles. This causes thermal energy to be given off. As energy cannot be created or destroyed it follows that the energy must have come from the energy that the ball possesses. Therefore of the GPE that the ball possessed at the beginning some energy is given off as thermal energy. This means that not all the GPE is converted into KE as it would have been if the ball had been dropped in a vacuum.
As the ball hits the floor with less KE than it would have done if it had been dropped in a vacuum it follows that less energy is converted into elastic potential energy and back into KE again. As the ball has less KE and is travelling slower it becomes stationary faster at the top of its arc. Therefore the height that it reaches is less high.
As velocity increases air resistance increases in proportion to the square of the velocity. This means that the faster that the ball travels the larger the force of air resistance upon it. Also the difference between the force of air resistance acting upon a ball travelling at 1ms-1 and the force of air resistance acting upon a ball travelling at 2ms-1 is far smaller than the difference between the force of air resistance acting upon a ball travelling at 20ms-1 and the force of air resistance acting upon a ball travelling at 21ms-1. The higher h1, the faster the velocity that the ball reaches. This means that the higher h1 the more h2 will differ from the height that the ball would have reached had it been dropped in a vacuum.
This proves that the higher h1 the more h2 will differ from the height that the ball would have reached had it been dropped in a vacuum. Also it shows inaccuracies in the experiment as it shows that heights were recorded that exceeded the height that the ball would have reached had it been dropped in a vacuum.
Analysis:
From the above table it can be seen that there were inaccuracies in the experiment. It shows that heights were recorded that exceeded the height that the ball would have reached had it been dropped in a vacuum. This is impossible. Either the coefficient to restitution that was worked out is incorrect, which would mean that the first three results are inaccurate or subsequent results were inaccurate. These inaccuracies could have been caused by external factors or parallax error even though efforts were made to avoid parallax error occurring - by dropping the ball one time that was not measured and placing a blob of blue tack onto the meter rule at the approximate height it bounced to. Then when dropping the ball again eye level was kept level with the blue tack. It was difficult to get down to the exact level of the blue tack seeing as it meant lowering your entire upper body in the short amount of time taken for the ball to hit the floor and rebound again to get your eye level from h1 to h2 (where the blue tack was stuck, approximately). For the higher heights the distance from h1 to h2 was almost a meter which meant it was difficult to get eye level from h1 to h2 to accurately in a short amount of time. For the lower heights the flight time of the ball was extremely short and again it was difficult to move ones head over the distance from h1 to h2 in order to obtain accurate results.
The procedure allowed me to observe the affects of the height from which a table tennis ball was dropped from upon the height to which it bounced. This was when the ball was falling in air. It provided me with five repeats so that the maximum and minimum results could be discounted and a reliable average could be taken. It was more reliable to use the middle three results as it automatically discounted any anomalies; assuming two similar anomalies were recorded for one height, if they were then they both would be discounted. However if two results are recorded that do not fit the trend of other results nor are close to results recorded for that height then the experiments accuracy would be brought into doubt. This did not happen in my experiment however.
Averages are more reliable than using one result as they take into account variation between results. Taking an average of several results creates a measurement in the middle of the variation created by the experiment, which is the result that is closest to the height that would be recorded for the ball’s bounce if it were measured in an experiment that was totally accurate.
Working out the variation in results shows how accurate the experiment was. A totally accurate experiment would have a variation between results equal to zero however an experiment that is totally accurate needs to be conducted under conditions where air pressure and temperature remained constant, error produced by the ball falling on different parts of the linoleum floor tile (which was not totally even and thus produces inaccurate results) and human error removed by dropping the ball onto a uniform surface and using machinery to record the height to which the ball bounced to. The equipment necessary to generate these conditions was not available and as a result the results obtained were not one hundred percent accurate.
The experiment was conducted well however as the utmost efforts were brought into place to avoid parallax error and it was ensured as far as possible that factors that affected how high the ball bounced, excluding the height, were kept constant throughout the experiment. Removing the maximum and minimum results and taking an average of the middle three results also provided more accurate results.
The maximum and minimum results were included when working out the variation between results however, seeing as the maximum and minimum results were produced by the experiment and are therefore part of the variation between results produced by the experiment.
Table showing Variation between Results
All of the factors that could have affected the results that were uncontrollable could have produced variations between results. However the conditions were kept the same for each drop of the ball. Changes in air pressure could have affected results as could changes in temperature however changes in these two factors would have been small; air pressure would not have changed enough to affect the results in the hour period in which the experiment was conducted, and although the room’s temperature may have increased by a degree or two, due to body heat, over the course of the period temperature was not a major factor that affected the height to which the ball bounced and would not have significantly affected the results. Changes in the area of linoleum floor tile that the ball collided with may have affected the height to which it bounced to thus producing variation between results.
The average variation between results was 3cm. The variation between results was obtained by taking the minimum result away from the maximum result. These two results were excluded when averages were being calculated and therefore the average variation between results used for calculating the average was even less than 3cm.
From this I am able to determine that the experiment was very accurate. Also the facts that there were no anomalies and that all of the points were very close to the line of best fit show that the experiment was relatively accurate.
The Graph on page 23 shows that all of the results were very close together. This also proves the accuracy of the experiment.
The results of the experiment were obtained with a method that ensured that every drop was under similar conditions which ensured a fair test. Therefore the results are valid.
Improvements that could be made to the experiment if future work was to be done:
- Carry out more repeats
- Do a larger amount of results; 1cm, 2cm, 3cm, 4cm… etc.
- To a wider range of results i.e. use four meter sticks and go right the way up to four meters. This would allow one to find the terminal velocity of the ball.
- Use a uniform surface to drop the ball onto
- Use two people to measure the results; one person to drop the ball and one to measure the height to which it reaches after bouncing. This would eliminate parallax error further. An alternative method would be the measuring person holding a video camera level with the approximate height that the ball reaches after bouncing and videoing the ball reach the top of its arc. This would mean that one could re-examine the height to which it bounced to and find it exactly instead of having to make a split second judgement which is not half as accurate.
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Measuring the height to which the ball bounced on subsequent bounces would be interesting, seeing if h3 = h2 × CR. and thus if h3 = h1 × CR2.
To provide additional relevant evidence I would conduct further work as follows;
I would like to conduct the same experiment in a vacuum. This would provide evidence on how the height from which the ball is dropped from affects the height to which it bounces without air resistance. This would allow the actual coefficient to restitution to be calculated.
Apparatus:
Vacuum pump, rigid plastic cylinder, two large rubber bungs to fit over the two ends of the plastic cylinder, table tennis ball, Two meter stick rulers.
The apparatus will be set up as shown:
The ball will then be dropped:
The ball bounces back up:
h1 = The distance between the bottom of the ball before it is dropped and the ground.
h2 = The distance between the bottom of the ball at the top of its arc after bouncing and the ground.
This experiment would provide me with more results that are relevant to the experiment that I have already conducted.
