Bouncing Ball Experiment: Relationship between Initial Height of a Drop and Fractional Loss of Kinetic Energy

An Experiment to investigate how the Initial Height of which a Ball is dropped affects its Fractional Loss of Kinetic Energy

Research Question:

What is the relationship between initial height of a drop and the fractional loss of kinetic energy in a bouncing ball?

Introduction:

When an object is held at a height above the ground, it possesses gravitational potential energy (Ep = mg∆h) that is directly dependent of the mass and height of which the object is positioned above the ground. When it is dropped, Ep is converted into kinetic energy (Ek = ½ mv2), energy the object now possesses due to its motion. An inelastic collision, where an object collides with another and springs apart, occurs when the object hits the ground. According to Newton’s 3rd law, both the ground and object exert an equal but opposite force on each other upon collision, but as the ground is more massive than the object; it is the object that is rebound. However, also upon collision, kinetic energy is transformed into other types of energy, such as heat, elastic and sound energy. Therefore, the object does not return to its initial height, and will lose more and more Ep and Ek with every bounce until it is stationary.

Aim:

The aim is to determine how the initial height of which a standard basketball (≈0.06kg) is dropped (0.20m, 0.40m, 0.60m, 0.80m, 1.00m, 1.20m, 1.40m, 1.60m, 1.80m and 2.00m) will affect the fractional loss of its kinetic energy in 3 bounces.

Hypothesis:

In this investigation, the initial height of which a basketball is dropped will be altered in order to increase or decrease the fractional loss of kinetic energy within 3 bounces. The ball will rebound higher if it is dropped from a higher height than a lower height, as a result of the amount of gravitational potential energy, dependent on height above ground, converting into kinetic energy during the fall. However, kinetic energy is lost upon collision. It is possible that the same amount of energy is lost each time due to factors such as the force of friction in relation to the texture of the floor and air resistance. Therefore, if the initial height is higher, there will be more kinetic energy retained compared to a lower initial height, where less kinetic energy will be retained.

Variables:

Independent Variable
The independent variable is the initial height of which the basketball is dropped that will be increased 10 times (including the first trial) by 0.20m. These lengths will be 0.20m, 0.40m, 0.60m, 0.80m, 1.00m, 1.20m, 1.40m, 1.60m, 1.80m and 2.00m; measured out using a 1 meter ruler (±0.02m). This will be done by positioning the bottom of the basketball at the highest height, 2.00m, and decreasing it to the next smaller length, 1.80m, 1.60m and so on. Each length will be tried 4 times and averaged.

Dependent Variable
The dependent variable is the basketball’s fractional loss of kinetic energy (Ek) within 3 bounces, observed by eyesight and recorded on video. Data will be taken from the bottom of the basketball. A standard basketball will be held at a height measured against 2 meter rulers (±0.02), which allows the calculation of its gravitational potential energy through the equation gravitational potential energy = Mass x Gravity x Height, or Ep = mg∆h. In the first bounce, final Ek should equal the initial Ep, (Ekf = Epi OR ½mv2 = mg∆h). Height of rebound will then be measured and recorded in the second and third bounce, which will be used to calculate its difference to the intial Ep, and therefore the amount of Ek lost during the collision. At this point 3 values have been calculated; Ekinitial, Ekrebound and Eklost, or Ei, h2 and Eklost. The fraction of Ek lost is then calculated by the formula, (Eki – Ekf)/Eki.
For example, m= 0.60kg h= 2.00m

Initial drop

Epi = mg∆h
Epi = 0.60kg*9.81ms-2*2.00m
Epi = 11.772 J = Ekf

Bounce 1

hr1= .8m
Ep = mg∆h
Ep = 0.60kg*9.81ms-2*0.80m
Ep1 = 4.7088 J = Ekr1
Ekf – Ekr1
11.772 J - 4.7088 J
Eklost= 7.0632 J
Eki = 11.772 J
Ekf = 4.7088 J
Ekl= 7.0632 J
Fraction of Ek lost = (Eki – Ekf)/Eki
Fraction of Ek lost = (Eklost)/Eki
Fraction of Ek lost = (11.772 J – 4.7088 J)/ 11.772 J
Fraction of Ek lost = 7.0632 J /11.772 J
Fraction of Ek lost = 0.6

Controlled Variables

Variable
How variable will be controlled and measured
Possible affect on data if not controlled
Type of surface
The experiment will be carried out in the same spot, so that the ball will strike the same type of tile surface every time. There is no specified uncertainty associated with the surface; the manufacturing of the tile will be trusted.
If not controlled, this could affect the amount of Ek lost upon collision with the floor as the amount of friction compared to one type of surface, i.e stone versus wood, would vary. This would mean different amounts of Ek being converted into heat energy.
Type of ball (size, mass, material)
The same basketball will be used for all trials. This will make sure mass, size, shape, volume and material will be kept consistent. Mass of the basketball will be measured using an electronic scale (±0.001g or ±0.00001kg).
Mass could vary if ball was not kept the same, and would affect the calculation of energy as E=mg∆h. Also, the type of material impacting on the ground could affect the force of friction acting on the ball.
# of bounces
Rebound height will only be taken after the first bounce. This information will be controlled by observation of frame by frame video capture and data recording through logger pro video analysis.
More or less energy may be lost with every bounce; i.e 0.30 loss in the first bounce but 0.20 lost in the second bounce.
Temperature
Temperature in room will be monitored and kept the same at room temperature (25˚C) by keeping the air conditioner at 25˚C.
If not controlled the ball could heat up or cool down, affecting the amount of initial energy it has in total, affecting results especially after multiple trials.
Position of where height and rebound height is measured on the ball
Height and rebound height will be measured from the same point on the ball, the bottom, every time.
The uncertainty in height and rebound height will be affected and skewed, therefore affecting the calculation for fractional loss of kinetic energy.

Apparatus:

1x standard basketball
2 x 1m ruler (±0.01m)
1 x video recorder
1x retort stand w/ clamp OR tripod
1 x electronic scale (±0.00001kg)
1x roll of masking tape

Method

Part A: Set Up

1. Use masking tape to join the 2 meter sticks together so that the total length that can be measured is 2m
2. Tape meter sticks flat to a wall, making sure that it is straight from the ground to the top, as to prevent and reduce parallax error
3. Set up retort stand or tripod with video recorder at a far enough distance, eye level so that space is left between the top and bottom of meter sticks and the floor/ceiling.
4. Measure mass of basketball using electronic scale and record in data under “Data Collection” heading
5. Press record on video recorder

Part B: Collect Data

6. Position ball at highest initial drop height (2.00m)
7. Drop ball and let bounce until it comes to a complete stop
8. Repeat steps 5-7 for different initial drop heights (1.80, 1.60, 1.40… 0.20) with 4 trials each.

Part C: Extraction of h2 by Logger Pro video analysis

9. Add video into project
10. Set origin by using the “set origin” function and clicking the bottom of the meter sticks
11. Set scale by using the “set scale” function – click top of meter sticks and drag to bottom of meter sticks
12. Find point in video right before drop
13. Drop data point at the bottom of the ball (h1)
14. Find frame in video where the ball has bounced one time and is static in the air (the highest height), right about to fall again (h2)
15. Record h1 and h2 data on table (see example table below method)
16. Repeat steps 9-15 for all trials

Part D: Calculation

17. Using h1 and h2 values in table, calculate initial energy before drop (Ei) using the equation Ei = mg∆h1 and final energy after drop (Ef) using Ef= mg∆h2.
18. Calculate energy lost by Ei - Ef.
19. Calculate fractional loss of kinetic energy (Ekf) using the formula Ekf=(Ei - Ef)/ Ei.
20. Repeat steps 17-19 for all trials
21. Calculate average in h2 and Ekf by (T1+T2+T3+T4)/4

Example table(s) for data collection:

Initial Height ,h1, ±0.02m
Trial #

Initial Energy before drop, Ei, ±0.10J Ei = mg∆h1
Rebound height, h2, ±0.02m
Final energy after rebound, Ef, ±0.10 J Ef = mg∆h2
Loss of Kinetic Energy, Eklost, ±0.20 J
Eklost=Ei – Ef
Fractional Loss of Kinetic Energy, Ek,, ±0.02 Ek,=(Ei–Ef)/Ei
0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

Initial Height, h1, ±0.02m
Average (T1+T2+T3+T4)/4

Rebound height, h2, ±(max-min)/2
Fractional Loss of Kinetic Energy, Ek, ±0.10 J
2.00

1.80

1.60

1.40

1.20

1.00

0.80

0.60

0.40

0.20

Data Collection:

m= 0.60453kg

Table(s):

Table 1: Table to show raw results and processed calculations for Fractional Loss of Kinetic Energy, Trial 1

Initial Height ,h1, ±0.02m
Trial 1

Initial Energy before drop, Ei, ±0.10J Ei = mg∆h1
Rebound height, h2, ±0.02m
Final energy after rebound, Ef, ±0.10 J Ef = mg∆h2
Loss of Kinetic Energy, Eklost, ±0.20 J
Eklost=Ei – Ef
Fractional Loss of Kinetic Energy, Ek,, ±0.02 Ek,=(Ei–Ef)/Ei
2.00
11.86
1.27
7.53
4.33
0.37
1.80
10.67
1.18
7.00
3.68
0.34
1.60
9.49
0.98
5.81
3.68
0.39
1.40
8.30
0.85
5.04
3.26
0.39
1.20
7.12
0.70
4.15
2.97
0.42
1.00
5.93
0.62
3.68
2.25
0.38
0.80
4.74
0.62
3.68
1.07
0.23
0.60
3.56
0.43
2.55
1.01
0.28
0.40
2.37
0.29
1.72
0.65
0.28
0.20
1.19
0.15
0.89
0.30
0.25

Table 2: Table to show raw results and processed calculations for Fractional Loss of Kinetic Energy, trial 2

Initial Height ,h1, ±0.02m
Trial 2

Initial Energy before drop, Ei, ±0.10 J Ei = mg∆h1
Rebound height, h2, ±0.02m
Final energy after rebound, Ef, ±0.10 J Ef = mg∆h2
Loss of Kinetic Energy, Eklost, ±0.20 J Eklost=Ei – Ef
Fractional Loss of Kinetic Energy, Ek,, ±0.02 Ek,=(Ei–Ef)/Ei
2.00
11.86
1.32
7.83
4.03
0.34
1.80
10.67
1.08
6.40
4.27
0.40
1.60
9.49
0.93
5.52
3.97
0.42
1.40
8.30
0.80
4.74
3.56
0.43
1.20
7.12
0.67
3.97
3.14
0.44
1.00
5.93
0.65
3.85
2.08
0.35
0.80
4.74
0.57
3.38
1.36
0.29
0.60
3.56
0.49
2.91
0.65
0.18
0.40
2.37
0.27
1.60
0.77
0.33
0.20
1.19
0.14
0.83
0.36
0.30

Table 3: Table to show raw results and processed calculations for Fractional Loss of Kinetic Energy, trial 3

Table 4: Table to show raw results and processed calculations for Fractional Loss of Kinetic Energy, trial 4

Initial Height ,h1, ±0.02m
Trial 4

Initial Energy before drop, Ei, ±0.10 J Ei = mg∆h1
Rebound height, h2, ±0.02m
Final energy after rebound, Ef, ±0.10 J Ef = mg∆h2
Loss of Kinetic Energy, Eklost, ±0.20 J Eklost=Ei – Ef
Fractional Loss of Kinetic Energy, Ek,, ±0.02 Ek,=(Ei–Ef)/Ei
2.00
11.86
1.31
7.77
4.09
0.35
1.80
10.67
1.10
6.53
4.14
0.39
1.60
9.49
0.96
5.69
3.80
0.40
1.40
8.30
0.83
4.92
3.38
0.41
1.20
7.12
0.70
4.15
2.97
0.42
1.00
5.93
0.61
3.62
2.31
0.39
0.80
4.74
0.58
3.44
1.30
0.28
0.60
3.56
0.45
2.67
0.89
0.25
0.40
2.37
0.28
1.66
0.71
0.30
0.20
1.19
0.13
0.77
0.42
0.35

Table 5: Table to show average processed calculations for Fractional Loss of Kinetic Energy in relation to initial drop and rebound heigh

Initial Height, h1, ±0.02m
Average (T1+T2+T3+T4)/4

Rebound height, h2, ±(max-min)/2
Fractional Loss of Kinetic Energy, Ek, ±0.10 J
2.00
1.31 ±0.03
0.35
1.80
1.09 ±0.01
0.39
1.60
0.95 ±0.03
0.41
1.40
0.83 ±0.03
0.41
1.20
0.69 ±0.02
0.43
1.00
0.62 ±0.03
0.38
0.80
0.58 ±0.01
0.28
0.60
0.44 ±0.01
0.27
0.40
0.28 ±0.01
0.30
0.20
0.14 ±0.01
0.31

Note: All crossed out / highlighted data points are outliers (1.80, trial 1, 0.80 trial 1, 0.60, trial 2) and have been disregarded in the calculation for the average data corresponding to its respective initial drop height.

Values in the columns titled ‘Initial Energy before drop, Ei’ were calculated using the formula Ei = mg∆h1, where mass m of the basketball was measured to be 0.60453±0.00001kg, g is generally accepted value for gravity 9.81ms-2, and h1 is height 0.20 – 2.00m, measured by observation upon drop and upon computerized data analysis though Logger Pro.

For example, the initial energy before drop at 2.00m, trial 1 was calculated as follows:

Ei = mg∆h1
Ei = 0.60453±0.00001kg *9.81ms-2*2.00±0.02m
Ei = 11.8608786
Ei= 11.86

∆y/y = ∆a/a + ∆b/b
∆Ei/Ei = ∆m/m + ∆h1/h1
∆Ei/11.8608786= 0.00001kg/0.60453kg + 0.02m/2.00m
(∆Ei/11.8608786)* 11.8608786= 0.0100165418*11.8608786
∆Ei= 0.118804986
∆Ei= ±0.12
∆Ei= ±0.10

∴ Ei = 11.86 ± 0.10 J with ≈ 0.84% ≈ 1% uncertainty

The values in the columns titled ‘Final Energy after drop, Ef’ were calculated using the same formula with replacement of h1 initial drop height to h2, rebound height, Ef=mg∆h2. The example below uses the h2 data from 2.00m, trial 1.

Ef = mg∆h2
Ef = 0.60453±0.00001kg *9.81ms-2*1.27±0.02m
Ef = 7.531657911 J
Ef = 7.53 J

∆y/y = ∆a/a + ∆b/b
∆Ef/Ef = ∆m/m + ∆h1/h1
∆Ei/7.531657911= 0.00001kg/0.60453kg + 0.02m/1.27m
(∆Ef/7.531657911)*7.531657911= 0.0157645733*7.531657911
∆Ef= 0.1187333732
∆Ef= ±0.12
∆Ef= ±0.10
∴ Ef = 7.5 ± 0.10 J ≈ 1.33% ≈ 1% uncertainty

‘Fractional loss of Kinetic Energy, Ekf’ is calculated using the formula Ekf= Ei-Ef/Ei. Although not used in this investigation, values in column Ekf can also be calculated using the formula 1- h2/h1. The column titled ‘Loss of Ek, Eklost’ was created to simplify the calculation of Ekf, as Eklost= Ei – Ef, part of the the Ekf formula. These values can be calculated as followed:

Eklost= Ei – Ef
Eklost= 11.86 J – 7.53 J
Eklost= 4.33 J

∆y= ∆a + ∆b
∆Eklost= ∆Ei + ∆ Ef
∆Eklost= 0.10+0.10
∆Eklost= ± 0.20
∴Eklost = 4.33 J ± 0.20 J ≈4.62% ≈ 5% uncertainty

Ekf= Ei-Ef / Ei
Ekf= Eklost / Ei
Ekf= = 4.33 J ± 0.24 J / 11.86 ±0.12
Ekf = 0.3650927487
Ekf =0.37

∆y/y = ∆a/a + ∆b/b
∆Ekf/Ekf= ∆Eklost /Eklost + ∆Ei/Ei
∆Ekf/0.3650927487= ∆0.24 /4.33 + ∆0.12/11.86
(∆Ekf/0.3650927487)*0.3650927487= (0.05542725173+ 0.010)* 0.3650927487
∆Ekf=0.02388701517
∆Ekf= ±0.02
∴Ekf = 0.37 ± 0.02 ≈ 5.41% ≈ 5% uncertainty

The averages in table 5 are calculated by adding all results associated with each initial drop height from all trails and dividing by 4; (T1+T2+T3+T4)/4. For example, the average fractional loss of kinetic energy for 2.00m was calculated as follows:

(T1+T2+T3+T4)/4
=(0.37+0.34+0.34+0.35)/4
= 0.35

However, as some data points were disregarded on the basis of being outliers, they were not taken into account for. Therefore in this case, the average was taken by adding all valid data points, i.e (T1+T2+T3)/3. For example, the average fractional loss of kinetic energy for 0.60m was calculated as follows:

(T1+T3+T4)/3
=(0.28+0.27+0.25)/3
=0.266666
= 0.27

The uncertainties for these values were calculated by (max value – min value)/2.

∆Average Ekf= (max value – min value)/2
∆Average Ekf= (0.44-0.25)/2
∆Average Ekf= ±0.095
∆Average Ekf= ±0.10

∆Average h2= (max value – min value)/2
∆Average h2= (0.45 – 0.43)/2
∆Average h2 = ±0.01m

Graph 1: Graph to show average Fractional Loss of Kinetic Energy is affected by initial drop height
Conclusion:

The results in graph 1 and table 5 show the average fractional loss of energy that correlate to the 4 trials of 10 changes to the initial drop height. Results demonstrate that increasing the initial drop height does not in fact affect the fractional loss of kinetic energy of the basketball. Instead, the average factional loss of kinetic energy stayed approximately 0.35 ±0.10 no matter what height it was dropped at. Results were slightly higher when the ball was dropped from a higher height and slightly lower at lower heights, but stayed within ±0.10 of the average value. This is most probably due to the greater force of friction from air resistance and surface type acting on the ball, converting the kinetic energy into heat and sound energy. Kinetic energy could also have been converted into elastic energy (energy to do with the disfiguration of volume or shape) upon impact. These forces have less of an impact when the ball is dropped from a lower height, as there is less displacement to travel through and therefore less air to resist. Friction from the ground should be constant as the ball only impacts for a very short period of time. Therefore this proves the hypothesis; that the same amount of energy is lost each time, to be true and reliable.

Hence, the linear, almost completely horizontal gradient (max, min and best) in graph 1 proves that fractional loss of kinetic energy stays consistent whether the initial drop height increases or decreases.

Evaluation:

Table 6: Table to show evaluation of uncertainties and errors in investigation (in order of significance)

Source of Uncertainty
Random/ Systematic
Possible Improvement
Extraction of h2 by Logger Pro video analysis – uncertainty of ‘set scale’ function is unknown and if the camera is not level but in fact at an angle. This is significant as it can affect the collection of correct h2, and therefore the experimental value of fractional loss of kinetic energy.
Random / parallax / systematic
Instead of, or along side video analysis, it would benefit to use a motion sensor to help plot a displacement vs. time graph. This could allow for more accurate collection of rebound height and therefore calculations for Ef and Ekf. This would also help confirm the values by having reproducible data.
Temperature – Trials for 0.20-0.60m were taken on a separate day from trials from 0.80-2.00m. Although room temperature was the same, the basketball was stored in a separate room prior to the trials and therefore could have been affected by different temperature or other environmental factors.
Random
Take all trials on the same day and make sure ball is stored at a constant environment prior to investigation. It would also benefit if temperature was further/more precisely controlled by cooling down / heating up the ball to a specific temperature after every trial.
Height, h1 (±0.02) – measured by observer dropping the ball. Due to the round spherical shape of the ball, it is difficult for the observer to very accurately position the basketball in the correct initial drop height. However, as the observer is doing this at eye level with the ball, it can be assumed that the error is small enough and can be considered negligible.
Systematic
If a flat surface with a clamp holding the ball (i.e, a machine much like the one in an arcade claw game) were to position and then let go of the ball at the correct drop height.
Mass, m
(±0.00001kg) – measured with an electronic scale (tarred before measurement) in g (±0.001g) and converted into kg by dividing by 1000. Error is small enough and can be considered negligible.
Systematic
Use more precise electric scale that measures to 4+ sig figs/ decimal places. Make sure calibration is set to zero.