Formal Lab Report #1
Date: October 28, 2011
David Guckenberger & Austin Kaiser
Summary
Two methods of experimentally quantifying the coefficient of restitution (e), for bouncy balls
contacting cement were compared. The coefficient of restitution (COR) was deduced by
releasing a ball from a known height and measuring the time between bounces. The more
accurate experiment, used a microphone and computer software to gauge the time between
bounces. eliminating human error through timing with a stopwatch. This switch to an automated
timing reduced the uncertainty by one order of magnitude, resulting in values of e = .863 ± 0.09
and e = .918 ± 0.008 , for the racket ball and golf ball, respectively.
Intro
Many sports use specifically designed balls that can only be utilized for their intended purpose.
For example, a tennis ball could not be used to correctly play basketball or golf. Each ball is
inherent to its own sport and features a distinct set of characteristics. One such characteristic is
ball-bounciness. Bounciness is defined as the amount a ball rebounds after contacting a surface,
provided it has some initial velocity. The bounciness is relative to a constant called the
coefficient of restitution (COR). The COR is a ratio relating the velocities from before and after
a bounce. Ratios of 1 and 0 define materials that are perfectly elastic and completely inelastic,
respectively. Presented herein are two techniques that were utilized to calculate the COR of a
ball contacting a given surface. In addition, the uncertainty of the measurements for each
technique has been calculated to determine the more effective method of calculating the COR.
Experimental Technique
To develop an effective technique for
measuring the COR of a ball, two
experiments were compared. In both cases,
the ball was rolled off a platform from a
height (h) of 5 ft. onto a concrete surface.
This height was measured using a tape
measure (Stanley model #32-279) attaining
an accuracy of 0.03125 in. The value used for
gravity (g) was 32.2 ft/s2, with no uncertainty
assigned. (figure 1) The first method of
gathering time utilized a manual stopwatch
(Accusplit model #S3MAGXLBK), with an
uncertainty of .001 s. However, because of
human error associated with starting and
Figure 1. Experimental technique schematic.
stopping the stopwatch when the ball hits the ground, the time uncertainty is on the order of .06
s. Therefore, method two was derived to eliminate human error. A microphone (Shure model
#SM58)(uncertainty ~.010 s.) was incorporated with LabVIEW (National Instruments, myDAQ)
to record the time between the bounces. The microphone detected the sound from bounces one
and two. Simultaneously, the software started a timer at bounce one and stopped it at bounce
two, giving a more accurate measure of time. Both methods were conducted using a golf ball
(Snake Eyes) and a racket ball (Penn Ultra Blue), to confirm that data between the two
experimental techniques were similar.
Data Reduction
In order to calculate the velocity of a falling ball, measurements of height and time were needed.
The resulting COR was calculated as the ratio of velocities before and after impact with the
ground. The COR(e) equation was derived from Newton’s Law and Conservation of Energy
Theory (Equation 1).
e=
𝑡𝑏𝑜𝑢𝑛𝑐𝑒
2
𝑔
√2ℎ
(1)
The uncertainty of the height measurement (uh) is one half of the resolution of the tape measure,
making uh = 1/32 in. The uncertainty of time is the product of the standard deviation of the time
and the student t value (tv,95)(equation 2).
𝑈𝑡 = 𝑡𝑣 ,95 ∗ 𝑠𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑣 = 𝑁 − 1
(2)
A 95% confidence interval was used for all uncertainty calculations, where 21 measured of
bounce time (N) were taken to reduce the overall uncertainty. The Root Sum Square method was
used for the propagations of uncertainty (equations 3 – 5).
𝜕𝑒
1
𝑔
Ue, t = 𝜕𝑡 * Ut = 2 *√2ℎ * Ut
𝜕𝑒
Ue, h = 𝜕ℎ * Uh =
−𝑡√𝑔
3
4(2ℎ) ⁄2
Ue = √𝑈𝑒,𝑡 2 + 𝑈𝑒,ℎ 2
* Uh
(3)
(4)
(5)
Measurements and calculations summarizing the data reduction results available in the Appendix
(table 1).
Results
The two methods of measuring COR demonstrated a drastic different of uncertainty. The COR
calculated for the racket ball and golf ball using the manual timer method were 0.87 ± 0.05 and
0.90 ± 0.06, respectively. Similarly, the COR calculated for the racket ball and golf ball using a
microphone was 0.863 ± 0.009 and 0.918 ± 0.008, respectively. The calculated COR is
comparable in both methods, and as expected, the COR of the golf ball is closer to one than that
of a racket ball, due to the elastic nature of the golf ball. However, the uncertainties of the two
methods differ by an order of magnitude. This difference is most likely due to the human error
inherent to manually timing with a stop watch, in addition to the delay between pressing the
button and the timer actually stopping.
To confirm the effectiveness of the technique, two different balls were used in both methods, and
the height was varied. The use of two balls, and visual verification of the COR similarity
between methods demonstrated the consistency between methods. To further verify that the
microphone method was effective, a golf ball was released from 6ft instead of 5ft, and the COR
came out to be 0.911 ± 0.011, which falls between the values calculated in the previous
methods. The results of the calculated COR for the microphone experiment are demonstrated in
the histograms (figure 2). Results for data determined by manually timing the bounces are
available in the appendix (figure 3 – A3).
Figure 2. COR frequency taken with the microphone
Appendix
A1: Actual Data for microphone measurements
Golf Ball
Freq
Dist
0.231
0.192
0.385
0.077
0.077
#
Norm
Hist
80
67
134
27
27
Norm
Dist
73
105
85
39
10
T (sec)
e
Bins
Freq
1
1.019
0.914
0.914
6
2
1.022
0.917
0.917
5
3
1.025
0.920
0.920
10
4
1.020
0.915
0.923
2
5
1.017
0.912
0.926
2
6
1.016
0.912
7
1.023
0.918
Statistics
8
1.018
0.913
9
1.021
0.916
t (sec)
e
10
1.022
0.917
min
1.016
0.912
11
1.022
0.917
max
1.032
0.926
12
1.024
0.919
ave
1.023
0.918
13
1.023
0.918
stdev
0.004
0.004
14
1.023
0.918
count
21
21
15
1.023
0.918
Bins
4.559
4.559
16
1.024
0.919
Bins
5
5
17
1.032
0.926
Bin Size
0.0032 0.002871
18
1.017
0.912
T_value
2.086
19
1.031
0.925
20
1.027
0.921
21
1.023
0.918
22
1.025
0.920
23
1.017
0.912
24
1.028
0.922
25
1.023
0.918
A2 – Uncertainty
Propagation
of microphone timed data:
26
1.029
0.923
MICROPHONE:
1. Racket Ball from height (h) = 5 ft.
Uh = 0.0625
Ue, t =
1
2
*√
Avg. time (t) = 0.96 s
C.O.R. (e) = 0.86
Ut = 2.086*St= 2.086 * 0.005= 0.010
32.2
2∗5
* 0.010 =.009
Ue = √0.0092 + (−0.0014)2 =
Ue, h =
.0091
−0.96√32.2
3
4(2∗5) ⁄2
* 0.03125 = -0.0014
2. Golf Ball from height (h) = 5 ft.
0.918
Uh = 0.0625
Ue, t =
1
2
*√
Avg. time (t) = 1.023 s
Ut = 2.086*St= 2.086 * .004= 0.009
32.2
2∗5
* 0.009 = 0.0081 Ue, h
=
−1.023√32.2
3
4(2∗5) ⁄2
Ue = √0.00812 + (−0.0014)2 = .0082
A3: Frequency of manually timed data
Figure 3. COR results from stopwatch experiment
* 0.003125 = -0.0014
C.O.R. (e) =