University of Colorado Boulder

ASEN 2003: Dynamics and Systems

Lab 2 - Section 013 - Group 16

Bouncing Ball Experiment Lab

Authors:Kelly Crombie ∗Isobel Griffin †Jarrod Puseman ‡Sevi Senavinin §

February 14th, 2018

∗SID: 106968138†SID: 105070884‡SID: 104003252§SID: 105178134

The purpose of this lab is to use knowledge of particle dynamics to derive and analyzeexpressions used for the coefficient of restitution, and to gain a physical understanding of directcentral impact collisions. Experimentally, a ball was dropped in comparison to a ruler, and theheight and time were recorded from the first to the third subsequent bounces. Three methodswere used to calculate the coefficient of restitution for a ping pong ball: height of bounce, timeof bounce, and time to stop. A golf ball was also tested using the height of bounce method. Onetrial used Logger Pro to determine the height of bounce. Overall, the golf ball had a highercoefficient of restitution than the ping pong ball. The most accurate method was the height ofbounce because it had a substantially lower error than the other two methods.

ContentsI Nomenclature 1

II Theory 1

III Experimental Procedure 2A Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3B Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3C Method 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3D Improved Method 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4E Golf Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4F Logger Pro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

IV Results 5

V Performance Analysis 6A Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

VI Conclusions and Recommendations 8

I. Nomenclature

e = Coefficient of restitutionh = Height, [m]t = Time, [s]m = Mass, [kg]v = Velocity, [m/s]g = Acceleration due to gravity, [m/s2]δx = Uncertainty in x, [m]

II. Theory

When a ball is dropped vertically onto a horizontal surface, the dynamics of the system can be modeled as a directcentral impact of a particle. If the ball is released from rest at an initial height h0, the height of subsequent bounces andthe total time the ball bounces may be used to estimate the coefficient of restitution of the ball and floor. The objectivesof this experimental lab are to derive the expressions for e as a function of ball height, time between bounces, andtime to stop bouncing. These values are measured several times for different balls, and then compared with derivedestimations of e. The final results are then assessed based upon experimental error. [2]

1

Fig. 1 Height vs sequential bounces

A ball is released from rest at a height, h0, above a horizontal surface. The height of the mass center of the ball aftern subsequent bounces is termed hn. The time required for a single bounce is tn, and for the ball to stop bouncing is ts .

Direct central collisions are defined as collinear impacts, where linear momentum is conserved, and the velocitiesbefore and after impact are related by the coefficient of restitution e. If e = 0, the collision is perfectly plastic, and if e =1, the total kinetic energy is conserved. [1] Given a ball falling vertically on a horizontal surface and the principles ofenergy and momentum for direct central collisions, the coefficient of restitution can be estimated from three differentmethods (equations 3-5). Each of these derivations begin with the general equations for the direct central collisions,

mAvA + mBvB = mAv′A + mBv′B (1)

where v′A and v′B are representative of the velocity of each object after collision. The coefficient of restitution is

defined as

e =v′B − v′AvA − vB

(2)

Estimation of eheight from height of bounces is given by

eheight =(

hnhn−1

) 12

=

(hnh0

) 12n

(3)

Estimation of ebounce from time of 2 adjacent bounces is given by

ebounce =(

tntn−1

)(4)

Estimation of estop from time to stop bouncing is given by

estop =ts −

√2h0g

ts +√

2h0g

(5)

The derivation of these equations for the coefficient of restitution is given in Appendix B.

III. Experimental ProcedureTo ensure accurate data collection, a experimental test area was setup to allow for multiple trials. See Fig. 2. The

experiment consisted of three methods. Each method is aimed at measuring the coefficient of restitution in a differentway. Method 1 calculates the coefficient of restitution by using the heights of successive bounces. Method 2 uses thetime between successive bounces. Last, method 3 uses the total time between release and the ball coming to rest. Eachmethod had a slightly different procedure which is discussed below. To ensure reliability between methods some factorswere held constant. In each trial, the ball was dropped from a height of 36 inches. Additionally, the ball was dropped inthe same location for each test. For tests involving height measurements, a grid system was setup and taped to the labcart in order to ensure it would not move between trials. The grid system was measured using the same tools to measureinitial height. Overall, 53 trials of gathering data were used to explore the coefficient of restitution for this lab.

2

A. Method 1This method calculates the coefficient of restitution by measuring the height of each bounce. Ten trials measured the

first, second, and third bounce heights of a ping pong ball dropped from 36 inches in height. In order to ensure accuratedata collection we followed a strict test setup and testing procedure. Our test setup consisted of a yard stick which wastaped down to prevent movement. Our test setup also included graph paper which was marked every inch. Duringtesting all footage was recorded using a slow motion camera. The testing procedure is as follows:

1) Ball dropper positioned notebook along the 36 in mark.2) Ball dropper positioned ball.3) Countdown. 3...2...1...Release!4) Camera begins filming on 1.5) Filming ceases after 3 bounces.6) Analyze film frame by frame to find the height of each bounce

Sources of error for this method include error in the measurement device and possible unexpected efficiencies in thebounce height itself. The graph paper was marked reasonably accurately with the one-inch gradings, but measuring theheight has some uncertainty. If the phone was at an angle, the ball would appear at a different height than it actuallywas. To minimize this effect, the phone was held in a plane parallel to the sheet and positioned close to the middle ofthe graph sheet. Also, bounce heights can be affected by the ball and bounce surface. Dirt particles on the floor candecrease the bounce height and even deflect the ball if large enough.

B. Method 2The second method consisted of recording the time of consecutive bounces. The time was measured from the

bottom of the first impact and again at successive impacts. The test setup remained the same as that in Method 1. Theprocedure is as follows:

1) Ball dropper positioned notebook along the 36 in mark.2) Ball dropper positioned ball.3) Countdown. 3...2...1...Release!4) On first impact timer begins time.5) On second and third bounce time is recorded.

Despite the rigorous method, there are a few errors. The biggest error is the delay between when the timer hears theball hit the ground and hits the lap button. Another error to consider is the misplacement of the ball as it may not alwaysbe dropped from exactly 36 inches.

C. Method 3The third method consisted of timing the ball until it came to rest. Again, the ball was dropped from a height of 36

inches while another person timed it. The test setup remained the same as in method 1. The procedure is as follows:

1) Ball dropper positioned notebook along the 36 in mark.2) Ball dropper positioned ball.3) Countdown. 3...2...1...Release!4) On release the timer starts recording time.5) When the ball has ceased moving in the Z direction the time is stopped.

While precautions were taken, there is still a lot of error in this method. The biggest source of error will be thedelay in human reaction between release and the start of timing. There is also going to be a delay when the ball stops.Another error is that sometimes time may be stopped prematurely. While there is no analytical numbers for these errorsthey should still be kept in mind.

3

D. Improved Method 3Due to Method 3 being the most simple, we decided it had the most potential for minimal error with improvements.

Since most error came from human reaction time, we decided to minimize the amount of people involved. Instead ofhaving a person release the ball and another person measure time, we had one person do both. We kept the test setup thesame as in methods 1,2, and 3 to prevent any differences to cause additional error. The testing procedure is as follows:

1) Ball dropper positioned notebook along the 36 in mark.2) Ball dropper positioned ball.3) Countdown. 3...2...1...Release!4) Upon "Release" Ball dropper drops ball and starts their timer.5) When the ball has ceased moving in the Z direction the time is stopped.

This would minimize the error in time between actual release of the ball and start of the timer, though there still existserror in the total time measurement and still a small error in starting the time.

E. Golf BallThe procedure used for the golf ball portion of the experiment was the exact same as Method 1. Please refer to

section III A. Method 1 for details.

F. Logger ProThe Logger Pro software was used on three drops to track the ball from recorded videos of the experiment. The

video was first converted into MP4 using VLC Media Player, then imported into Logger Pro. Video Analysis in theprogram is used.

1) Click set origin, then select the top of the ball.2) Click set scale, click and drag over a known distance in the frame and set to that distance.3) Click add point, select the top of the ball again. Video advances 1 frame.4) X and Y Position data is recorded over time.5) Once sufficient data is collected, export file as text file.

Error in this method is largely due to issues such as camera angle and measurement accuracy of Logger Pro. Logger Prosimply uses the pixels in a frame to make distance calculations, so the edges of the frame will have some distortion, andany movement in the camera also causes the frame to shift artificially. Though timing error is brought down since theball can be analyzed frame by frame, the error in height is about the same as that of Method 1.

4

(a) Measuring Total Time The ball was droppedaway from the test station while the height gage wasbeing created. A horizontal notebook is used tomark the plane at 36 inches.

(b) Measuring Bounce Heights and Times Theball is dropped against marked graph paper as abackground. The drops are filmed with a phone,and the data are retrieved by reviewing the videoby eye.

Fig. 2 Ball Drop Test Setups

IV. Results

Fig. 3 Coefficient of restitution vs test number

5

Table 1 Experimental Measurements, Regular Ping Pong Drop

Method 1Rebound 1 [± 1 in]

Method 1Rebound 2 [± 1 in]

Method 2Bounce 1 [± 0.2 s]

Method 2Bounce 2 [± 0.2 s]

Method 3Total Time [± 0.3 s]

1 27.8 21.0 0.65 0.62 9.322 28.1 21.1 0.71 0.58 8.763 27.0 20.0 0.76 0.64 8.464 28.0 20.2 0.69 0.66 9.305 28.9 21.0 0.79 0.64 8.856 28.2 21.2 0.75 0.66 9.307 28.1 20.6 0.72 0.64 8.708 28.2 21.3 0.84 0.62 8.319 28.6 21.0 0.73 0.63 7.7110 27.9 20.6 0.73 0.57 8.58

Table 2 Experimental Measurements, Non-standard Ping Pong

Improved Method 3Total Time [± 0.3 s]

Method 1Golf Ball

Height 1 [± 1 in]

Method 1Golf Ball

Height 2 [± 1 in]

1 8.90 30.5 25.252 8.80 29.75 24.253 9.15 30.75 25.254 8.20 30.75 25.05 8.18 31.0 25.56 8.75 31.0 25.57 8.53 30.75 25.58 9.1 30.25 25.59 8.73 30.25 25.2510 8.9 30.25 25.0

Table 3 Experimental Measurements, Processed with Logger Pro

Logger ProHeight 1 [±0.0254 m]

Logger ProHeight 2 [±0.0254 m]

Logger ProTime Bounce 1 [±0.05 s]

Logger ProTime Bounce 2 [±0.05 s]

1 0.6191 0.4602 0.7330 0.60002 0.5966 0.3475 0.7000 0.55603 0.6191 0.4602 0.7330 0.6000

V. Performance Analysis

After running all the trials through MATLAB, our team was able to calculate multiple values for the coefficient ofrestitution. In Table 3, shown below, one can see the results of the first three trials, as well as the improved method, golfball, and Logger Pro. For method 1, a value of 0.9 was found using our experimental method; but, when analyzed withLogger Pro a value of 0.796 was found. For method 2, a value of 0.86 and 0.815 were found which again were relativelyclose. For method 3, a value of 0.9 and 0.82 were found. The difference in values was caused by the improvement of the

6

method. While there was a discrepancy between the Logger Pro results and the experimental, it is due to the accuracy ofthe method. When you compare the improved method 3 it is very close to the Logger Pro values. This is because theLogger Pro values are also an improved method, in a sense. Logger Pro allowed for closer analysis of the heights andtime between bounces. Since the results of the Logger Pro and the improved method 3 align, it is safe to assume theactual value for e is around 0.81.

A. Error AnalysisTo ensure meaningful data collection, error analysis was performed on all calculations. Using principles learned in

ASEN 2012 at the University of Colorado, Boulder; the General Formula for error analysis was applied. The generalformula ultimately takes the error in each portion of the calculation and gives a net value at the end. For example,there could be an error in initial height, height of bounce 1, height of bounce 2, and height of bounce 3; the generalmethod takes the error in all 4 measurements and finds how the error propagates into the final calculation. The genericformula is shown below where X and Y are two sources of possible error. The general formula provides a more accurateestimation of error than other formulas because it accounts for error in each variable in the experiment.

δ =

√( ∂ f∂x

δX)2+

( ∂ f∂y

δY)2

(6)

For Method 1, the coefficient of restitution is given by equation (3):

eheight =(

hnhn−1

) 12

=

(hnh0

) 12n

Applying the general method to this yields

δe,height =

√√√[1

2hn−1

(hn

hn−1

) −12

δhn

]2+

[−hn

2(hn−1)2

(hn

hn−1

) −12

δhn

]2(7)

Recall that the coefficient of restitution for the second method is given by equation (4)

ebounce =(

tntn−1

)Applying the general formula for error propagation gives

δe,bounce =

√(1

tn−1δtn

)2+

(tn(tn−1)2

δtn−1

)2(8)

For the third method, start with equation (5)

estop =ts −

√2h0g

ts +√

2h0g

In the same manner as above, the error in this calculation for the coefficient of restitution is found with

δe,stop =

√√√√√√√√√( 2√ 2h0g(ts +

√2h0g

)2 δts)2 + ( −√2ts√h0g

(√2h0g + ts

)2∗ g

δh0

)2(9)

The values and error are then plugged into all these equations to find the uncertainty in the various methods forcalculating the coefficient for restitution.

7

Table 4 Coefficients of Restitution

Ping PongMethod 1

Ping PingMethod 2

Ping PongMethod 3

Ping PongImproved Method 3

Ping PingLogger ProMethod 1

Ping PingLogger ProMethod 2

Golf BallMethod 1

0.900 ± 4 ∗ 10−6 0.860 ± 0.2571 0.900 ±0.003 0.82 ± 0.002 0.796 ± 4 ∗ 10−6 0.815 ± 0.142 0.920 ± 4 ∗ 10−6

VI. Conclusions and Recommendations

Overall, computing the errors in this lab was the most challenging and tedious part. Recording and timing thedifferent trials was extremely simple. It was exciting to watch the ball bounce and see all the different errors in real life.For example, in one trial we noticed the ball bounced off a piece of dirt on the ground which radically changed the ballsvelocity. Being able to visualize the errors made the calculations more interesting and real. Group 16 learned a greatdeal about the importance of calculating error. All of the results came in at fairly similar values for the coefficient ofrestitution; however, the errors showed which methods were most accurate. The method that proved the most accurate incalculating the coefficient of restitution was using the height in between two consecutive bounces. As shown in Table[4], Method 1 consistently had the lowest error values that were all 4e-6. The worst was Method 2, using the time ittook the ball to completely stop bouncing. The errors in this method ranged from 0.1 to 0.26 (Table [4]) which aresubstantially greater than the errors from the other two methods. The golf ball had a higher coefficient of restitution thanthe ping pong ball.

When determining the coefficient of restitution, accuracy is extremely important. One recommendation from Group6 is for teams to use a brightly colored ping pong ball when using the Logger Pro system to find height and time data. InGroup 16’s experiment, a white ping pong ball was used against a white backdrop of grid paper, thus it was difficult todetermine the exact location of the ping pong ball in each frame. Another recommendation is to use a better camerathat can record at slow motion. This would make the Logger Pro portion of the lab much more simple. Logger Promoves frame by frame, but the ball is moving so quickly that it is hard to determine its exact location in each frame.This uncertainty in location resulted in high uncertainty in the calculations for coefficient of restitution. Using a slowmotion camera would hopefully make the location of the ball in each frame far more clear which in turn would result inextremely accurate data and thus calculations.

8

References

[1] Hibbeler, R. C. Engineering Mechanics: Dynamics. 5th Ed. Upper Saddle River, NJ: Pearson/Prentise Hall,2004. Print.

[2] McMahon, Jay. ASEN 2003 Lab 2: Bouncing Ball Experiment. Canvas. University of Colorado Boulder, Web.Jan. 2018.

Acknowledgments

Group 16 would like to thank Teaching Assistant Trevor Fritz for his assistance in determining how to best drop theball in each trial. Group 16 would also like to thank the ITLL staff for providing the materials necessary to perform eachexperiment.

Appendix A: Contributions

Kelly Crombie: Raw Data analysis code for E values. Write Up: Method 1,2,3, Improved 3 and Performance analysisIsobel Griffin: Logger Pro analysis and LaTeX formattingJarrod Puseman: Derivations and all LaTeX tablesSevi Senavinin: Raw data analysis, error analysis partials

Appendix B: Derivations

Below are the steps to derive the equations used for calculating the coefficient of restitution for each method. Forevery case, start at the definition of the coefficient of restitution:

e =V′

A − V′B

VB − VA(10)

We note that for these purposes, the floor is considered to have zero velocity before the impact since the referenceframe is based on the earth, and the floor will have negligible velocity after the impact since the mass of the ping pongball is so small compared to the earth. Considering these, equation (10) becomes

e =V′

A

VA(11)

This can be rewritten substituting Vn for the nth new velocity V′

A as

Vn = e ∗ Vn−1 = e ∗(e ∗ Vn−2

)= e2 ∗ Vn−2 = ... = en ∗ V0 (12)

where V0 is the very initial velocity with which the ball strikes the ground.

Method 1To get the equation used in method 1, one must find the velocity of the ball as a function of height alone. If we

ignore air drag and friction from the floor to the ball, see that the velocity of the ball at a given time t after launch is:

V = V0 − g ∗ t (13)

At the peak of a bounce, the velocity of the ball will be instantaneously 0 at a point:

V = V0 − g ∗ t = 0V0 = g ∗ t

t =V0g

(14)

9

Also recall that in the case of constant acceleration,

h = h0 + V0 ∗ t +a2∗ t2 (15)

For us, the h0 is 0, and the acceleration is the acceleration due to gravity so that

h = V0 ∗ t −g

2∗ t2 (16)

We can substitute the time of the maximum height we solved in (14) into (16) to get

h = V0 ∗V0g− g

2∗

(V0g

)2=

V20g−

V202g

=V20g

(17)

Solving for V0, we get the initial vertical launch velocity for any given height.

h =V20g

V20 = g ∗ hV0 =

√g ∗ h (18)

Since we are ignoring drag effects, we assume the ball returns to the ground with the same velocity it launches fromat the previous bounce. Thus

Vn =√g ∗ hn , Vn−1 =

√g ∗ hn−1 (19)

This allows the claim that

e =Vn

Vn−1

=

√g ∗ hn√g ∗ hn−1

=

(g ∗ hng ∗ hn−1

) 12

e =

(hn

hn−1

) 12

(20)

Method 2The equation for method 2 is derived similarly. Start with equation (16):

h = V0 ∗ t −g

2∗ t2

We can solve for V0 as a function of the total time of a bounce by setting the height equal to 0.

0 = V0 ∗ t −g

2∗ t2

0 = t

(V0 −

g

2t

)0 = V0 −

g

2t

V0 =g

2t (21)

10

This is the initial vertical velocity of any projectile path as a function of the time required to return back to theground. See that

e =V′

A

VA

e =g2 tn

g2 tn−1

e =(

tntn−1

)(22)

Method 3To get the equation used in Method 3, we start with equations (12) and (21).

VA =g

2t

Vn = en ∗ V0

The total time for the ball to stop bouncing then can be written as an infinite series after the initial drop

ts = tdrop + tbounces

ts =V0g+

2 ∗ e ∗ V0g

+2 ∗ e2 ∗ V0

g+

2 ∗ e3 ∗ V0g

+ ...

ts =V0g+

∞∑i=1

2eiV0g

(23)

The sum of an infinite geometric series is given by

S∞ =a1

1 − r (24)

For us, this is

S∞ =2eV0g

1 − e (25)

11

so that the total time is given by equations (23) and (25)

ts =V0g+

2eV0g

1 − e

(1 − e) ∗ ts = (1 − e)V0g+

2eV0g

(1 − e)(ts −V0g) = 2eV0

g

1 − ee∗ (ts −

V0g) = 2V0

g

(1e− 1) ∗ (ts −

V0g) = 2V0

g

1e− 1 =

2V0g

(ts − V0g )1e− 1 = 2V0

tsg − V01e=

2V0tsg − V0

+ 1

1e=

2V0tsg − V0

+tsg − V0tsg − V0

1e=

V0 + tsgtsg − V0

e =tsg − V0tsg + V0

(26)

Note that V0 is the velocity of the ball when it first hits the ground and is found with the conservation of energy

mgh0 =12

mV20

V0 =√

2gh0 (27)

Substituting (27) into (26), we get

e =tsg −

√2gh0

tsg +√

2gh0

e =ts −

√2h0g

ts +√

2h0g

(28)

12

Appendix C: MATLAB Code1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 % ASEN 2003 : Bouncing Ba l l Expe r imen t Lab3 %4 % This s c r i p t does c a l c u l a t e s t h e c o e f f i c i e n t o f r e s t i t u t i o n u s i ng t h r e e5 % d i f f e r e n t methods , u s i n g e x p e r im e n t a l d a t a .6 %7 % Group 16 Members : Ke l l y Crombie [106−968−138]8 % I s o b e l G r i f f i n [105−070−884]9 % Ja r r o d Puseman [104−003−252]

10 % Sev i S en av i n i n [105−178−134]11 %12 % Crea t ed : 01 / 31 / 2018 − Kel l y Crombie13 % Modi f i ed : 02 / 07 / 2018 − Mit Sen av i n i n14 %15 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%16

17 %% Housekeeping18 c l o s e a l l ; c l e a r ; c l c ;19 s e t ( 0 , ’ d e f a u l t t e x t I n t e r p r e t e r ’ , ’ l a t e x ’ ) ; % LaTex f o rm a t t i n g20

21 %% De f i n i t i o n s22 h i = 1 ; % I n i t i a l h e i gh t , [m]23 g = 9 . 8 0 7 ; % Ac c e l e r a t i o n due t o g r a v i t y , [m/ s2 ]24

25 %% Data Hardcoded26

27 Ts = [ 6 . 6 8 7 . 22 8 . 86 8 . 05 7 .33 8 .55 7 .78 8 .42 8 .25 9 . 3 2 ] ;28 Ts2 = [ 8 . 7 6 8 . 49 9 . 85 8 .85 9 .30 8 .06 8 .63 8 .31 8 .17 8 .58 8 . 9 8 ] ;29

30 bounce = . . .31 [ 27 . 8 21 . 0 1 5 . 5 ; . . . % Drop 132 28 .1 21 .1 1 6 . 1 ; . . . % Drop 233 27 .0 20 .0 1 5 . 1 ; . . . % Drop 334 28 .0 20 .2 1 5 . 2 ; . . . % Drop 435 28 .9 21 .0 1 4 . 8 ; . . . % Drop 536 28 .2 21 .2 1 6 . 2 ; . . . % Drop 6 4 t h bounce 12 .5 i n ch37 28 .1 20 .6 1 5 . 5 ; . . . % Drop 738 28 .2 21 .3 1 6 . 5 ; . . . % Drop 839 28 .6 21 .0 1 4 . 8 ; . . . % Drop 940 27 .9 20 .6 1 5 . 6 ; . . . % Drop 1041 ] . ∗ ( 2 . 5 4 / 1 0 0 ) ; % Conve r s i on t o me t r i c42 [ row co l ] = s i z e ( bounce ) ;43

44 %% Impor t t h e d a t a45 [ ~ , ~ , raw ] = x l s r e a d ( ’ Expe r imen t a lDa t a . x l s x ’ , ’ Shee t1 ’ , ’B3 : I13 ’ ) ;46

47 %Cre a t e o u t p u t v a r i a b l e48 d a t a = r e s h a p e ( [ raw { : } ] , s i z e ( raw ) ) ;49

50 %Cre a t e t a b l e51 Expe r imen t a lDa t a = t a b l e ;52 %Al l o c a t e impo r t ed a r r a y t o column v a r i a b l e names

13

53 Expe r imen t a lDa t a . Method = d a t a ( : , 1 ) ;54 Expe r imen t a lDa t a . VarName3 = d a t a ( : , 2 ) ;55 Expe r imen t a lDa t a . VarName4 = d a t a ( : , 3 ) ;56 Expe r imen t a lDa t a . VarName5 = d a t a ( : , 4 ) ;57 Expe r imen t a lDa t a . VarName6 = d a t a ( : , 5 ) ;58 Expe r imen t a lDa t a . Improved = d a t a ( : , 6 ) ;59 Expe r imen t a lDa t a . G o l f b a l l = d a t a ( : , 7 ) ;60 Expe r imen t a lDa t a . VarName9 = d a t a ( : , 8 ) ;61

62 T r i a l 2 = t a b l e 2 a r r a y ( Expe r imen t a lDa t a ( : , 3 : 4 ) ) ;63

64 bounce2 = t a b l e 2 a r r a y ( [ Expe r imen t a lDa t a ( 1 : 1 0 , 7 ) Expe r imen t a lDa t a ( 1 : 1 0 , 8 ) ] ). ∗ ( 2 . 5 4 / 1 0 0 ) ;

65

66 Ts3 = t a b l e 2 a r r a y ( Expe r imen t a lDa t a ( : , 6 ) ) ;67

68 [ r2 c2 ] = s i z e ( T r i a l 2 ) ;69 e r r o r = Expe r imen t a lDa t a ( 1 1 , : ) ;70

71 %% Ca l c u l a t i n g e based on t h e h e i g h t o f t h e bounce72 % fo r j = 1 : row73 % fo r i = 2 : c o l74 %75 % eh1 ( j , i ) = bounce ( j , i ) / ( bounce ( j , i −1) ) ^ . 5 ;76 %77 % end78 % end79 %eh1avg = ( eh1 ( : , 2 ) + eh1 ( : , 3 ) ) / 2 ; % Equa t i on on l e f t s i d e80

81 f o r j = 1 : row82 f o r i = 1 : c o l83 eh2 ( j , i ) = ( bounce ( j , i ) ) ^ ( 1 / ( 2 ∗ c o l ) ) ;84 end85 end86 eh2avg = ( eh2 ( : , 1 ) + eh2 ( : , 2 ) + eh2 ( : , 3 ) ) / 3 ;87 eh2avga = sum ( eh2avg ) / l e n g t h ( eh2avg ) ;88

89 %% Ca l c u l a t i n g t h e e based on t h e t ime of Bounce90 f o r j = 1 : r291

92 eb ( j , 1 ) = T r i a l 2 ( j , 2 ) / T r i a l 2 ( j , 1 ) ;93

94 end95 ebavg= sum ( eb ) / l e n g t h ( eb ) ;96 %% Ca l c u l a t i n g t h e e based on t h e t ime t o s t o p97

98 f o r i = 1 : l e n g t h ( Ts2 )99

100 es ( i , 1 ) =( Ts2 ( i ) − s q r t ( ( 2∗ h i ) / g ) ) / ( Ts2 ( i ) + s q r t ( ( 2∗ h i ) / g ) ) ;101

102 end103 esavg= sum ( es ) / l e n g t h ( e s ) ;104

105 x= l i n s p a c e ( 1 , 1 1 , 1 1 ) ;

14

106 x1= l i n s p a c e ( 1 , 1 0 , 1 0 ) ;107

108 % p l o t t i n g109

110 f i g u r e ( )111 ho ld on112 g r i d minor113 p l o t ( x , eb , x , es , x1 , eh2avg , ’ L inew id th ’ , 1 . 5 )114

115 t i t l e ( ’Modulus o f R e s t i t u t i o n C o e f f i c i e n t ’ , ’ Fon tS i z e ’ , 16 )116 x l a b e l ( ’ Te s t Number ’ , ’ Fon tS i z e ’ , 14)117 y l a b e l ( ’ C o e f f i c i e n t o f R e s t i t u t i o n , $e$ ’ , ’ Fon tS i z e ’ , 14)118 l e g end ({ ’E t ime of bounce ’ , ’E t ime t o s t o p ’ , ’E h e i g h t o f bounce ’ } , . . .119 ’ l o c a t i o n ’ , ’ n o r t hwe s t ’ , ’ I n t e r p r e t e r ’ , ’ l a t e x ’ , ’ Fon tS i z e ’ , 12 ) ;120 s e t ( gca , ’ T i c k L a b e l I n t e r p r e t e r ’ , ’ l a t e x ’ ) ;121 %save a s ( f i g u r e ( 1 ) , ’ EPlo t ’ , ’ . jpg ’ ) ;122

123 %%Second Se t o f T r i a l s124 %Golf B a l l He igh t o f Bounce125 [ row co l ] = s i z e ( bounce2 ) ;126

127 f o r j = 1 : row128 f o r i = 1 : c o l129 eh3 ( j , i ) = ( bounce2 ( j , i ) ) ^ ( 1 / ( 2 ∗ c o l ) ) ;130 end131 end132

133 eh3avg = ( eh3 ( : , 1 ) + eh3 ( : , 2 ) ) / 2 ;134 eh3avga = sum ( eh3avg ) / row ;135

136 %Improved t ime t o s t o p137 f o r i = 1 : l e n g t h ( Ts3 )138 es2 ( i , 1 ) =( Ts3 ( i ) − s q r t ( ( 2∗ h i ) / g ) ) / ( Ts2 ( i ) + s q r t ( ( 2∗ h i ) / g ) ) ;139 end140 es2avg= sum ( es2 ) / l e n g t h ( es2 ) ;141

142 %% Er r o r Ana l y s i s143

144 h0 = 1 ; % I n i t i a l h e i gh t , [m]145 hn = 0 ; % He igh t a t number o f t r i a l146 n = 0 ; % Number o f t r i a l s , s t a r t i n g from 0147 dhn = 1∗ ( 2 . 5 4 / 1 0 0 ) ; % Un c e r t a i n t y i n h e i g h t a t n , [m]148 dh0 = 1∗ ( 2 . 5 4 / 1 0 0 ) ; % Un c e r t a i n t y i n what ?149 t n = 0 ;150 t n1 = 0 ;151 t s = 0 ;152

153 % Method 1154 [ row co l ] = s i z e ( bounce ) ;155

156 f o r j = 1 : row157 f o r i = 1 : c o l158

159 n = i ;

15

160 hn = bounce ( j , i ) ;161

162 ehn ( j , i ) = ( ( 1 . / ( 2 . ∗ n .∗ h0 ) ) .∗ ( hn . / h0 ) . ^ ( ( 1 . / ( 2 . ∗ n ) ) −1) .∗ dhn ) . ^ 2 ;163 eh0 ( j , i ) = (( − hn . / ( 2 . ∗ h0 . ^ 2 ) ) .∗ ( hn . / h0 ) . ^ ( ( 1 . / ( 2 . ∗ n ) ) −1) .∗ dh0 ) . ^ 2 ;164

165 EM1( j , i ) = s q r t ( ( ( eh0 ( j , i ) ^2 ) ∗ ( ( dh0 ) ^2 ) ) + ( ( eh0 ( j , i ) ^2 ) ∗ ( dhn ^2 ) ) ) ;166

167 ehng ( j , i ) = ( ( 1 . / ( 2 . ∗ n .∗ h0 ) ) .∗ ( hn . / h0 ) . ^ ( ( 1 . / ( 2 . ∗ n ) ) −1) .∗ dhn ) . ^ 2 ;168 eh0g ( j , i ) = (( − hn . / ( 2 . ∗ h0 . ^ 2 ) ) .∗ ( hn . / h0 ) . ^ ( ( 1 . / ( 2 . ∗ n ) ) −1) .∗ dh0 ) . ^ 2 ;169

170 EM1g( j , i ) = s q r t ( ( ( eh0g ( j , i ) ^2 ) ∗ ( ( dh0 ) ^2 ) ) + ( ( eh0g ( j , i ) ^2 ) ∗ ( dhn ^2 ) ) ) ;171

172 end173 EM1( j , 4 ) = sum (EM1( j , 1 : 3 ) ) / 3 ;174 EM1g( j , 4 ) = sum (EM1g( j , 1 : 3 ) ) / 3 ;175 end176 EM1avg = sum (EM1( : , 4 ) ) / row ;177 EM1gavg = sum (EM1g ( : , 4 ) ) / row ;178

179 % Method 2180 d t = . 2 ;181 [ rowt c o l t ] = s i z e ( T r i a l 2 ) ;182 f o r j = 1 : rowt183

184 f o r i = 2185 %Ping Pong Ba l l186 t n = T r i a l 2 ( j , i ) ;187 t n1 = T r i a l 2 ( j , i −1) ;188

189 e t n = 1 / t n ;190

191 e t n1 = − t n / ( t n1 ^2 ) ;192 EM2 = s q r t ( ( ( e t n ^2 ) ∗ ( d t ) ^2 ) + ( ( e t n1 ^2 ) ∗ ( d t ) ^2 ) ) ;193 end194

195 EM2( j , 1 ) = s q r t ( ( ( e t n ^2 ) ∗ ( d t ) ^2 ) + ( ( e t n1 ^2 ) ∗ ( d t ) ^2 ) ) ;196 end197 EM2avg = sum (EM2 ( : ) ) / l e n g t h (EM2) ;198

199

200 % Method 3201 f o r i = 1 : l e n g t h ( Ts )202

203 h0 = 1 ;204 t s = Ts ( i ) ;205

206 %Ping Pong Ba l l207 e t s = −((− s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) . / ( s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) . ^ 2 ) . . .208 + (1 / ( s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) ) ;209

210 eh0 = −(− s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) . / ( s q r t ( 2 ) ∗ g ∗ s q r t ( h0 / g ) ∗ ( s q r t ( 2 ) . . .211 ∗ s q r t ( h0 / g ) + t s ) . ^ 2 ) − ( 1 / ( s q r t ( 2 ) ∗ g ∗ s q r t ( h0 / g ) ∗ ( s q r t ( 2 ) . . .212 ∗ s q r t ( h0 / g ) + t s ) ) ) ;213

16

214 EM3( i , 1 ) = s q r t ( ( e t s ^2 ) ∗ ( d t ^2 ) + ( eh0 ^2 ) ∗ ( dh0 ) ^2 ) ;215

216 %Golf B a l l217 t s = Ts3 ( i ) ;218 e t s g = −((− s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) . / ( s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) . ^ 2 ) . . .219 + (1 / ( s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) ) ;220

221 eh0g = −(− s q r t ( 2 ) ∗ s q r t ( h0 / g ) + t s ) . / ( s q r t ( 2 ) ∗ g ∗ s q r t ( h0 / g ) ∗ ( s q r t ( 2 ) . . .222 ∗ s q r t ( h0 / g ) + t s ) . ^ 2 ) − ( 1 / ( s q r t ( 2 ) ∗ g ∗ s q r t ( h0 / g ) ∗ ( s q r t ( 2 ) . . .223 ∗ s q r t ( h0 / g ) + t s ) ) ) ;224

225 EM3G( i , 1 ) = s q r t ( ( e t s g ^2 ) ∗ ( d t ^2 ) + ( eh0g ^2 ) ∗ ( dh0 ) ^2 ) ;226

227 end228 EM3avg = sum (EM3 ( : ) ) / l e n g t h (EM3) ;229 EM3gavg = sum (EM3G( : ) ) / l e n g t h (EM3G) ;230

231 %% P r i n t R e s u l t s232

233 f p r i n t f ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ r ’ )234 f p r i n t f ( ’ e based on h e i g h t i s %.2 f w/ e r r o r o f %.3 f \ r ’ , eh2avga , EM1avg ) ;235 f p r i n t f ( ’ e based on t ime of bounce i s %.2 f w/ e r r o r o f %.3 f \ r ’ , ebavg , EM2avg ) ;236 f p r i n t f ( ’ e based on t ime t o s t o p i s %.2 f w/ e r r o r o f %.3 f \ r ’ , esavg , EM3avg ) ;237 f p r i n t f ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ r ’ )238 f p r i n t f ( ’−−−−−−−−−−−−−Second Se t o f T r i a l s −−−−−−−−−−−−−−−\ r ’ )239 f p r i n t f ( ’ e g o l f b a l l based on h e i g h t o f bounce i s %.2 f w/ e r r o r o f %.3 f \ r ’ ,

eh3avga , EM1gavg ) ;240 f p r i n t f ( ’ e based on t ime t o s t o p i s %.2 f w/ e r r o r o f %.3 f \ r ’ , es2avg , EM3gavg ) ;

1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 % ASEN 2003 : Bouncing Ba l l Expe r imen t Lab3 %4 % This s c r i p t e s t i m a t e s t h e c o e f f i c i e n t o f r e s t i t u t i o n u s i ng t ime between5 % bounces and maximum h e i g h t be tween bounces (2 bounces ) .6 %7 % Group 16 Members : Ke l l y Crombie [106−968−138]8 % I s o b e l G r i f f i n [105−070−884]9 % Ja r r o d Puseman [104−003−252]

10 % Sev i S en av i n i n [105−178−134]11 %12 % Crea t ed : 02 / 05 / 2018 − I s o b e l G r i f f i n13 % Modi f i ed : 02 / 07 / 2018 − Mit Sen av i n i n14 %15 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%16

17 %% Housekeep ing18 c l o s e a l l ; c l e a r ; c l c ;19

20 %% Read i n d a t a from Logger Pro21

22 % T r i a l 123 A = r e a d t a b l e ( ’ v ideo1 −3. t x t ’ , ’ D e l im i t e r ’ , ’ \ t ’ , ’ Heade rL ines ’ , 1 ) ;24 A = t a b l e 2 a r r a y (A) ;25

17

26 % T r i a l 227 B = r e a d t a b l e ( ’ v ideo1 −3. t x t ’ , ’ D e l im i t e r ’ , ’ \ t ’ , ’ Heade rL ines ’ , 1 ) ;28 B = t a b l e 2 a r r a y (B) ;29

30 %T r i a l 331 C = r e a d t a b l e ( ’ v ideo1 −3. t x t ’ , ’ D e l im i t e r ’ , ’ \ t ’ , ’ Heade rL ines ’ , 1 ) ;32 C = t a b l e 2 a r r a y (C) ;33

34 %% T r i a l 1 Data Ana l y s i s35

36 % hardcode i n t ime i n t e r v a l s be tween bounces37 At_n_1 = A( 37 , 1 ) − A(15 , 1 ) ; %seconds38 At_n = A( end , 1 ) − A(37 , 1 ) ; %seconds39

40 At ( : ) = [ At_n_1 At_n ] ;41

42 % hardcode i n max h e i g h t s between bounces43 Ahmax = max (A( : , 2 ) ) ; %me t e r s44 Ah_n_1 = A( 15 , 2 ) − A(26 , 2 ) ; %me t e r s45 Ah_n = A( 37 , 2 ) − A(47 , 2 ) ; %me t e r s46

47 Ah ( : ) = [ Ah_n_1 Ah_n ] ;48

49 %% T r i a l 2 Data Ana l y s i s50

51 % hardcode i n t ime i n t e r v a l s be tween bounces52 Bt_n_1 = B( 3 8 , 1 ) − B(17 , 1 ) ; %seconds53 Bt_n = B( end , 1 ) − B(38 , 1 ) ; %seconds54

55 Bt ( : ) = [ Bt_n_1 Bt_n ] ;56

57 % hardcode i n max h e i g h t s between bounces58 Bhmax = max (B ( : , 2 ) ) ; %me t e r s59 Bh_n_1 = B( 1 5 , 2 ) − B(27 , 2 ) ; %me t e r s60 Bh_n = B( 3 8 , 2 ) − B(48 , 2 ) ; %me t e r s61

62 Bh ( : ) = [ Bh_n_1 Bh_n ] ;63

64 %% T r i a l 3 Data Ana l y s i s65

66 % hardcode i n t ime i n t e r v a l s be tween bounces67 Ct_n_1 = C( 3 7 , 1 ) − C(15 , 1 ) ; %seconds68 Ct_n = C( end , 1 ) − C(37 , 1 ) ; %seconds69

70 Ct = [ Ct_n_1 Ct_n ] ;71 % hardcode i n max h e i g h t s between bounces72 Chmax = max (C ( : , 2 ) ) ; %me t e r s73 Ch_n_1 = C( 1 5 , 2 ) − C(26 , 2 ) ; %me t e r s74 Ch_n = C( 3 7 , 2 ) − C(47 , 2 ) ; %me t e r s75

76 Ch = [ Ch_n_1 Ch_n ] ;77

78 %% Es t ima t e o f e _ h e i g h t f o r each d a t a s e t79

18

80 Ae_he igh t = ( Ah_n / Ah_n_1 ) ^ ( 1 / 2 ) ;81 Be_he igh t = ( Bh_n / Bh_n_1 ) ^ ( 1 / 2 ) ;82 Ce_he igh t = ( Bh_n / Bh_n_1 ) ^ ( 1 / 2 ) ;83

84 e _ h e i g h t = mean ( [ Ae_he ight , Be_he igh t , Ce_he igh t ] ) ;85

86 %% Es t ima t e o f e_bounce f o r each d a t a s e t87

88 Ae_bounce = At_n / At_n_1 ;89 Be_bounce = Bt_n / Bt_n_1 ;90 Ce_bounce = Ct_n / Ct_n_1 ;91

92 e_bounce = mean ( [ Ae_bounce , Be_bounce , Ce_bounce ] ) ;93

94 %% F i n a l o u t p u t t e d r e s u l t s95

96 f p r i n t f ( ’ e h e i g h t i s %.3 f \ n ’ , e _ h e i g h t ) ;97 f p r i n t f ( ’ e bounce i s %.3 f \ n ’ , e_bounce ) ;98

99 % END

19

NomenclatureTheoryExperimental ProcedureMethod 1Method 2Method 3Improved Method 3Golf BallLogger Pro

ResultsPerformance AnalysisError Analysis

Conclusions and Recommendations