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# Bouncing Ball Experiment

Extracts from this document...

Introduction

Alex Boorman        Ph20         -  -

Aim: To find out what affects the height to which a ball bounces.

Variables:

Height from which the ball is dropped

Mass of the ball

External factors, i.e. changing air density, temperature

The surface onto which the ball falls

Prediction – reasons for variable control:

Variables that will be altered:

The height the ball is dropped from will affect the height the ball bounces to due to the energy chain the ball goes through as it is dropped and bounces up again. The energy chain is as follows:

Therefore as energy cannot be created or destroyed the energy the ball starts with must be directly proportional to the energy the ball finishes with, at the top of its bounce, and so if the ball starts with more energy it must therefore finish with more. As in both cases the main form of energy is GPE it follows that the higher the ball is dropped from, the higher the ball will bounce.

• The ball starts with more GPE
• As there is more GPE more energy is converted into KE (The ball is going faster, KE=1/2mv2, v is greater therefore KE is greater by a larger amount). More thermal energy is also produced.
• As there is more KE, more energy is converted into elastic potential energy
• As there is more elastic potential energy, more energy is converted back into KE energy
• As there is more KE energy, more work would need to act upon the ball in order to stop it in the same distance. As gravity remains the same the amount of work acting on the ball remains the same (apart from slightly more air resistance due to the ball travelling faster and so hitting more air particles per second, but the effect of this is negligible) and the ball travels further before stopping. Therefore the ball goes higher.
• The higher the ball goes, the more GPE it ends up with. Therefore the ball ends up with more GPE

In short:

GPE=Mass (kg) × Gravitational Field Strength (N/Kg) × Height (m)

If we let mass = m

Gravitational Field Strength = g

Height = h1

An increase in h1, assuming g and m stay constant, results in an increase in    m × g × h1 which results in an increase in GPE.

A decrease in h1, assuming g and m stay constant, results in a decrease in    m × g × h1 which results in a decrease in GPE.

Therefore any change to the height from which the ball starts with affects the height to which it bounces to.

This is correct unless the ball shows signs of reaching terminal velocity.

The drag force increases as the ball goes faster. This is because as the ball goes faster it passes through a greater volume of air each second. It therefore hits more air particles each second and so the force of drag is bigger the faster the ball goes.

As the ball is accelerating due to gravity, at 9.81m/s2 it is constantly getting faster and therefore the drag force gets bigger and bigger. Drag is a squared function of velocity and therefore as the ball drops drag increases a greater amount each second. Once the drag force equals the gravitational force all forces are equal and acceleration stops. The ball has reached its terminal velocity and cannot fall any faster (unless dropped in a vacuum).

If a ball reaches terminal velocity at 20cm from the floor when dropped from 2m, it will reach terminal velocity at 10cm from the floor when dropped from 1.9m. This means it will be travelling the same speed when it hits the ground. This means that KE is the same for both balls when each ball hits the ground.

If the KE is the same as they hit the ground the energy stored in the ball as elastic potential energy will be the same also. If the elastic potential energy is the same then the same amount of energy is converted back into KE and so the balls leave the floor at the same speed. The balls leave the floor at the same speed with the same amount of KE and so both balls reach the same height and end up with the same amount of GPE at the top of their bounces.

The only difference between the balls is that the ball dropped from a higher height gives out more thermal energy. This is because the ball starts with more GPE. The balls finish up with the same amount of energy and the only energy given out is thermal energy. After the ball reaches terminal velocity, no more GPE is converted into KE is the ball cannot get any faster. Instead, as the ball is still always loosing GPE so long as it is still falling, all GPE is converted into thermal energy. Therefore the ball dropped from the higher height must give out more thermal energy in order to end up with the same amount of energy as the other ball.

Therefore the height the ball bounces will be proportional to the height that the ball is dropped from up to a certain point, where the ball begins to show signs of reaching its terminal velocity before it reaches the ground. Above this point the height all balls will bounce to will not be directly proportional to the height they are dropped from, but the increase in the height they bounce to will increase more slowly in proportion to the height they are dropped from compared to the increase between lower heights before the ball shows signs of approaching its terminal velocity before it hits the ground. I do not believe that any ball will reach terminal velocity in this experiment seeing as the maximum height that they can be dropped from is 2m and, as the ball is quite smooth, I do not believe that it will have enough time to accelerate to its terminal velocity before it hits the floor.

Prediction curve:

Variables that will be kept constant:

N.B:

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.

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

• If a ball is dropped in a vacuum. The ball starts at height h1. GPE = m × h1 × g
• Energy ball starts with = mh1g
• 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
• The proportion of energy lost when ball hits the floor = The Coefficient to the restitution of the two objects (CR)
• Energy ball leaves the floor with = CR (mh1g)
• All of the energy that the ball leaves the floor with is converted back into GPE
• GPE = CR (mh1g)
• At h2 the ball has energy CR (mh1g) or mh2g
• 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

Middle

2 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:

 Height the ball bounced to (cm) (h2) Height the ball was dropped from (cm) (h1) repeat 1 repeat 2 repeat 3 repeat 4 repeat 5 Average of all repeats Average of the middle three repeats 200 105 103 109 104 106 105.4 105.0 190 104 103 104 102 103 103.2 103.3 180 102 102 101 104 101 102.0 101.7 170 100 100 98 99 99 99.2 99.3 160 95 96 95 97 94 95.4 95.3 150 91 90 89 93 90 90.6 90.3 140 82 85 86 86 84 84.6 85.0 130 81 82 81 79 81 80.8 81.0 120 76 77 75 77 78 76.6 76.7 110 72 77 73 72 72 73.2 72.3 100 67 68 67 66 68 67.2 67.3 90 63 62 63 64 62 62.8 62.7 80 56 57 58 56 56 56.6 56.3 70 50 48 50 51 50 49.8 50.0 60 44 46 45 44 43 44.4 44.3 50 38 42 40 38 38 39.2 38.7 40 30 31 30 32 31 30.8 30.7 30 23 21 22 22 23 22.2 22.3 20 14 14 15 13 14 14.0 14.0 10 6 7 6 7 8 6.8 6.7

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.

Conclusion

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 ontoUse 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. 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.

Alex Boorman        Ph20        -  -

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