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147

Appendix A • A model to predict the trajectories of golf balls hitwith different clubs

The following program calculates the tanding velocity, angle to the vertical and spin of agolf ball projected from the ground with a velocity angle and spin introduced manually by[he operator. The model estimates the drag and Tift coefficients of the ball frominformation gained from wind tunnel tests on a Titleist golf ball by the AcushnetCompany. Thur, me calculations performed can only be estimates since the informationin the drag and lift calculations are for a single type of golf ball in a range of velocities.The model may, therefore, not be suitable for trajectories in which the initial backspin isextremely large or in which the angle of ascent is large. Information from the programshould not be used without the consent of Dr. A. J. Cochran.

PRINT "This program calculates the trajectory of a spinning golf ball"PRINT "COPYRIGHT Ar.Ccchran. revisions 1985,1986"PRINT "Lift and drag supplied confidentially by Acushnet Co .• New Bedford. USA"PRINTFOR Je l TO 200K,JNEXT JPRINT "ENTER [NITIAL SPEED (Ips), spin (rpm). Angle (degrees)"INPUT R,N.POCLS:PRlNT "The program can deal with headwind (-) or tailwind (..•.)"PRINT "ENTER WlNDSPEED [N fps (+/.)"INPUT WINDCLS:PRINT "The program uses a calculation steplength of O.lsec, prints trajectorydetails at intervals of I second and terminates after 10 seconds if the ball has not returnedto the ground by then."PRINT "00 you wish to use different values for any of these?"PRINT "Answer (Y)es or (N)o"INPUT ANSWERSIF ANSWER$,"N" THEN GOTO 280PRINT "Enter calculation step length, print interval and termination time you wish (inseconds)"INPUT H.H[.T!GOT0290280 H,O.I:HI=I:T!,IO290 PRINT "If drag and lift are standard please type 0 otherwise type J"INPUT Q2IF Q2,0 THEN GOTO 420PRIl'r'T "Enter the factors by which drag and lift are to be multiplied (e.g. 1.1,0.9)"INPUT CI.C2GOTOS60420C[=[,O.2GOTOSSO440 PRINT "Launch conditions"PRINT "SPEED ="R"fps; SPIN ,"N" rpm: ANGLE = "PO" degrees; WIND.'WIND" fps"(;OT0610SSO IF Q2,0 THEN S80560 CLS:PRINT "'DRAG ="Cl"times standard: LlfT="C2"timcs standard"GOTO S90S80 PRt"'T "DRAG and LIFT standard"590 PRINT "Calculation step length '"'"H" soc"600 GOTO 440610 READ M.GO,W.AREAD AO,AI,A2.A3READ 80,8[,82,82T .O:X=O, Y =0

[48

PO=PO' ATN(I)/4SU=R'COS(po)-WL'lDY=R'SIN(PO)PRINT" T(sec) X(ft) y(ft) U(fps) Y(fps)"1=01=0FOR T=OTO T!-II STEP II1=1+1R=SQR(U'U+ Y'Y)REM·"'The next twenty or so steps give functional approximations to DRAG and LlFfmeasured for a particualr Titleist ballDl,."AO+AI*R+A2·R*R+A3*R*R*R~DI+(N-IS70)'-')()()()()7)'(I-EXP(-.03'R))D=D*Clt.i ",BO+B 1*R+B2·R*R .•.B3*R *R "'RL-"Ll+ (.OOO244)'R '(I-EXP(- W'(N·1570»)L=L*C2F=GO'(·D'U·L'Y)I(M'R)G=GO'(-D'U+L'V)i(M'R)-GOU5:U+,S*H*FV5= V +.S*H*ONS=N'EXP(=A'Hi2)R5=SQR(US'US+YS'Y5)D5=AO+AI*R5+A2*R5·R5+A3*Rj*R5*RjDS=D5+(N5·1570)' .000007)'( I·EXP( -.03'RS))D5=D5'<:1L5= 80+ B I*R5+B2* RS*RS+B3*R5*R5 *R5W=L6+(.OOO244 )'RS'( I·EXP(· W'(NS·1570)))L~=W'C2F5=GO'(·DS'US-W'YS)/(M'RS)GS=GO'(·DS'U5+W'YS)I(M'RS)-GOUI=U+II'FSVl=V+H*GSX9=X:Y9=YX""X+.S*H·(U+U 1)Y=Y+.S*H*(V+Vl)U9=U:Y9=VT9=TU=UI:V=YlN=N'EXP(·A'II)P2=V·V9IF P2<O THEN1380IIIOQI=OIF Y>=O THEN 11S0QI=IGOTO 1320I1S0 IF I'H<1'1I1 THEN 12801160 J=J+IDIM 0(4)11700(O)=INT(lOO'(T'Hl+.5)II00O(I)=INT(IOO·(X+(T·II)·WIND)+.5)IIOO0(2)=INT(IOO'Y+.5)/I000(3)=INT(IOO·(U+W[ND)-.S)/IOO0(3)=INT(IOO·(U+W[ND)-.S)/IOOFOR K=OT04

@%=13IS94PRINT;O(K);

NEXT KIF QI=[ THEN 1400[270 RESTORE

[49

1280 NEXTT1290 DATA 0.10125.32.2,,000636 •.05DATA 0•.OOOU)(10OO3,2.7E-9DATA -.212,.0044,-.000025.5.28E-8X=(X*Y9-X9'Y)f(Y9- YlT=-H+«(T+Hj*Y9-T9'Y1/(Y9-Y)Y=OU=(U*Y9-U9*Y)/(U9- YlV=(V'Y9-V9'Yj!(V9-YlGOTO 11701380 Y7=Y9+.S·H·V9'2.'(V9-V)GOTO 11101400 A9::4S"'ATN(·V/(U+W1ND»fATN( I)V7"SQR«U.WINDjA2.V'Vjr.~:~~::~~~1~n!~rr#(1~1~~~~;;~;;':d~~:~::fps"PRINT "Spin = "INT(Nj"'l'm"END

1.10

Appendix 8 • Errors involved in the analysis or the photographsor ball impacts

B.1. Errors in the method using points representing the co-ordinates ofthe ball

(i) Errors of measurementThe smallest squares on the grid on Figure 2.10 are Smm x 5mm and the horizontal andvertical lines are marked at Imm intervals. It was estimated that a single co-ordinate hadan error of t: 1mm since this was the greatest accuracy possible. Figure B. t shows thepropagation of an error £ through the calculation of the velocity in equation 2.1, assumingthat the error in all the co-ordinates is the same. An initial error in placing a co-ordinate of1mm produces a final error of 2mm. The minimum distance between images of theincoming ball was found to be approximately 70mm and therefore. the greatest errorproduced in calculating the initial velocity of the ball was about 3%. The speed of theoutgoing ball was calculated by measuring the distance across a number of images; 6 or 7images usually measured about 7Omm. The error in this measurement is the same as thatfor the incoming ball SO that the percentage error was again about 3%.

Expression Accumulated errorh.• •h2, k2 2.h. 2••h2 +k2 .( (4.'(h' •• '».(h' +.') E

TABLE B.l. The propagation of an error IE through equation 4.1. An initial error ofImm produces a final error of 2mm. The greatest error in calculating the incomingvelocity was found to be 3% while trust for the outgoing velocity was often less than 1%.

An error in reading the co-ordinates of the ball was also present throughout thecalculations of the angle of arrival and departure of the ball (equations 2.3 and 2.4).Whcn two co-ordinates with an initial error of £ arc subtracted the combined error is~ F;v2 (Figure B.l). Thus. a maximum and minimum of Sj and e( were calculated andthis gave a range of possible angles. These were interpreted as an error t d9j and idge in8j and Sf respectively. Thus.

and.

dec = t x ( efm.u • efmin)

: ~ 1. (TAN" (V, - V, + • .(2) . TAN-! (V, - Y, + • .(2» eqn. D.2:r (X,. X,· e .(2) (X,· X,· • .(2)

If the distance between two images is about 70mm and the ball is travelling at 45° to thehorizontal. then an error of Imm in each of the co-ordinates gives a range of 43.36° to46.64° for the calculation of the angle. This is represented as an error of t 1.64°.

151

(ii) Systematic errors

Figure 8.1 (a) shows a diagram of a series of images of the ball 1000m apart and withthe grid misplaced at an angle CT to the true impact plane. Due to the angled grid thedistance between images appears as d' to the left of the centre of the photograph and as d"to the right of the centre of the photograph. d' and d" can be estimated as.

d' ;:::OA' + A' B::: dccstc-) + AA'tan(cr + 11-'):::dcosto-) ~ dsin(cr)lan(cr + \II)eqn. B.3

and.d" z 00 _DC' = _d __ ~ z _d __ d<jls;n(<jI)

costc-) costc-) cos(cr) cos(u)

B Displacedplane of grid

d' ~ deos(O'") + dsin(C7)tan(\V + 0"')

d" = ~) - ~~~rg.r)

FIGURE B. t (a). The effect of placing the grid accidentally at an angle (T to the trueimpact plane, The distance d would appear to be d' to the right of the centre of thephotograph and d" to the left of the centre of the photograph. The equations gi veapproximate calculations for the apparent distances between images, some estimates ofwhich are given in Figure C.l (b).

0' (') d'(mm) d" (mm)

1 100.3 99.7

5 102.3 98.6

10 106.1 97.1

FlGURE RI(b). Some estimates of the apparentdistances d' and d" as the angle between me scaleand the true plane of impact increases for an actualdistance between impacts of 100mm.

152

eqn. 8.4

Equations 8.3 and 8.4 are used to give some estimates of d' and d" as (T increases inFigure B. J (b). These were calculated for an initial distance between images of 100mm.At angles of 10° the error in d' is 6.1mm longer than the true value and d'' is 2.9mmshorter than the true value. It was possible to see errors of this magnitude when placingthe grid since it required misplacing both ends of the scale by about 45mm ~more thanone golf ball diameter. It was estimated that systematic errors of this sort were kept toless than 3%, requiring the scale to have been misplaced by less man half a golf balldiameter on either side of the true impact plane.

Figure B.2 (a) shows me effect of moving the grid a distance a to a plane parallel to thatof the true plane of impact The distance d is modified by :!: atan(o.:) and the apparentlengths of d' and d" with increasing a are shown in Figure 8.2 (b) if the actual distancebetween images is 100mm. The error reaches a value of 5% when i) is approximately30mm, This is one and a half times the radius of a golf ball and it was estimated that thescale could be placed well within this distance either side of the pitch marks.

,.-----------r'I----.,----r-...,.-- .•.Grid a furtheraway from plane

d'

Camera ~

FlGURE 8.2 (a). The effect of misplacing the grid nearer or further away than the trueplane of impact.

8 (mm) d' (mml d"(mm)

10 101.7 98.3

20 103.3 96.7

30 105.0 95.0

FIGURE 8.2 (b). The variation of the appdistances d' and d" as the distance a of thefrom the true plane of impact increases.

153

Appendix C • The calculation of spin In three dimensions usingtwo dimensional images

C.l. IntroductionDuring the analysis of the pictures using the co-ordinate method described in Appendix Babove it was considered that estimates of the side spin would help eo explain odd resultsthat occurred. The method described in this appendix was developed to calculate the threedimensional spin of the ball using the images on the two dimensional photographs. Itconsists of three stages. the first modifies the two dimensional co-ordinates from thephotographs to points in three-dimensional space. The second calculates the spin axis ofthe ball and the third stage calculates the magnitude of the spin and hence the spin in thethree dimensions.

C.2. Relief displacementThe relief displacement is the shift in the position of an image on a photograph caused bythe height of the object above a selected datum The relief displacement, with respect 10adatum, is outwards from the centre of the photograph (the principal point) for pointswhose elevations are above the datum. As an example, consider Figure C.I which showsa schematic diagram of an aerial view of a set of office buildings. Consider a co-ordinatesystem that is at ground level and that we wish to determine the position of the buildings.Since the bases of the buildings can not be seen on the photograph, the co-ordinates of thetops of the buildings have to be used. The building at the principal point is seen straighton and the co-ordinates of the top of this building are the same as those at the ground.However, an office block. away from the centre of the photograph is seen from an angleand the point representing the top of the building appears further from the principal pointthan irs base. It is necessary to reduce the distance R by a small amount 8 to ensure thatthe co-ordinate we measure is correct

R

Principalpoint

FIGURE C.I. A schematic diagram showing an aerial view ofa set of office buildingswith a co-ordinate system at ground level. If the co-ordinates of the tops of the buildingsare used 10 pinpoint the position of the buildings then the buildings to the right wouldappear too far to the right and the co-ordinates would have to modified by a small amount8.

This modification can also be used to alter the photographs of the golf ball impacts wherethe co-ordinate system is in the plane of the impact and the "tops of the buildings" arcpoints on the balls surface away from this plane (Figure C.2).

The modification of the two dimensional co-ordinates to three dimensional ones and thealteration for relief displacement takes place as follows. From the photograph of animpact (Figure 2.10 for example) the Y and Z co-ordinates of a point are measured. TheX co-ordinate is calculated assuming that the radius of the ball is 2L3m.m using,

x = .f (21.32• (Y - yo~2- f7.,-7.c}2)

154

eqn. C.l

The distance R is the distance, as measured on the photograph. from the principal point tothe point P (Figure C.2), It can be shown (Wolf, 1974) that R, X and ~ are related by,

z

ox

FlGURE C,2. A schematic diagram of a golfbaU whose centre lies in a plane containingthe principal point (0,0,0). The distance from the principal point to the point P ismodified by a distance! to the new point P' .

• RxXv "----r-

where 1. is the distance of the camera from the impact plane. The triangles OPA andOP'A' in Figure 3.8 arc similar and therefore we have the equauons,

8which, rearranged, give dY == Y(1 . In and dZ •• Z(J - K), where K = (l - If)' Thus wehave equations for the modified set of co-ordinates which are,

Y'.y - Y(I . K) = YK eqn. C.4Z' = Z - Z(I- K) = ZK eqn. C.5X· = v'(21.3' - (Y' - YO) - (Z' - Zo» eqn. C.6

WhcrcK"(1-~) eqn.C.7As an example, consider a photograph of a ball where the principal point is at CO, 0, 0),the centre of the ball is at (0,250,0) and a point on its surface is at (X, 264, 14). Thisrepresents a ball 250mm (0 the right of the centre of the photograph (as read on the scaleon the picture) with a point 14mm up and 14mm to the right of the centre of the ball. TheX co-ordinate is,

x = .,r (21.3' - (264 - 250) - (14 - 0» =.,r (453.69 - 392) = .,r (61.69) = 7.85mm.

The distance R would can be calculated as .,f ly2 + Z2) ""..{(2642 +- 142) ""264.37mm. Ifthe camera is approximately 600mm from the plane of impact then, using equation C.2,the relief displacement is,

8 - 264.3ki 7.85 =3.46mmandK' (1- 2~4~7) = 0.9869

Using equations 3.9. the modified co-ordinates are,Y' = 260.55Z' = 13.82

and X' c .,r(21.3'- (260.55 - 250)' - (13.82 - 0)'> = 12.30.The original co-ordinates of (7.85. 264, 14) have been modified for their reliefdisplacement to (12.3, 260.55.13.82). It should be noted that this modification was not

155

eqn. C.2

eqn. C.3

required when using the co-ordinates for the calculations in Appendix B since the pointsused lay in the impact plane where X = 0 and hence ~ = O.C.3. MelhodOnce the co-ordinates have been modified the)' can be used to calculate the spin axis of theball and hence the spin in the three dimensions. The vectors fl' (1 and (3 in Figure C.3represent a point on the surface of a ball, with respect to its centre, at times TI. Tl and T)respectively. The direction cosines of the vectors are written as,

COSQ:\=~,COSI3I:: ~,COS"'I:: ~

COSQ:2:::~' COSQ:2::~' COSQ:2••~

CO$0(3:::~' C(XOr:3",,~. COSO(;):: ~

eqn. e.g

eqn. e.9

eqn. e.IO

z

x

FIGURE C.3. The vectors (I> r1 and (3 are representations (with respect to the centre ofthe ball) of a point on its surface at times TI. T2 and T]. The direction cosines of thevectors can be used to calculate the spin axis of the ball and hence the spin in the threedimensions.

These equations do noe uniquely define the spin axis since the t sign indicates a choice ofspin direction, If the direction of rotation is known before any calculations are made thenthe signs of cosec, cos13 and cos j- are known. Once the spin axis is found it isrelatively simple to find the magnitude of the spin and hence the components .in eachdirection. Figure C.4 shows poinu PI and P:zat times TI and T2.

The vectors rJ. r1 and raare equidistant from the spin axis since they represent the samepoint on the surface of the ball at different times. Their magnitudes are the same and areequal to the radius of the ball. Using this fact. it can be shown that the direction cosinesof the spin axis are,

where,

ccsce =!,( CCl. cosl3 =! 8,{CC). cosy = ! ,{(I - C( 1+ 8'» eqa. e.11

156

A '"" (cos)" t - cosy 2)

(COS Y I • cosy 3)

B ~ {(COSO:: 1· COSO: 2) • A (cosOc: I - C050::3)}

{A(cos~l· COS~)J· (CO'~I· cos~,J)and

eqn. c.n

eqn. C.13

c = (cosy, - cosy I)

{(1· •.B!)(cosy 2' cosY ,,2 + (Iccsce I . C050::1) + 8 (C05(3 t . COS(32»)2}

cqn. C.14

Since the points are both equidistant from the spin axis then the angles between thevectors representing PI and Pz and the spin axis arc the same and can be calculated usingthe identity,

ccsa = ccsceccsee 1+ cos(3 cos{3J + cosy cos'j" I

cose = cOSOt:cOS0t:2 + cosp COS132+ cosy COSY2Since•

. I ¥P,.PI) ~sm~l'OI = ~ =2rsin(9)

then.

eqn. C.15 (a)

eqn. C.15 (b)

eqn. C.16

F'''--------.v

xFIGURE C.4. The points PI and PI represent a poinl on the ball at times TI and T2 andare equidistant from the spin axis. Using simple geometry the rotation ~rol can becalculated and, using the spin axis direction cosines, the spin in the three dimensions canbe found.

157

111isis the angle rotated through in the lime [fz - T,). Multiplying this by each of the spinaxis direction cosines (cosce , C05(3. cosy) give the components of spin in the threedimensions.

As an example, Figure C.5 shows a sphere of unit radius at times T" T2 and T). The ballis rotating in a clockwise direction and the co-ordinates of a point on the surface of thesphere at successive time intervals are (O.707, 0.5, 0.5), (0.707. 0.5, -0.5) and (0.707,-0.5. ·0.5).

(0.707. 0.5. 0.5) (0.707.0.5. -0.5) (0.707. -0.5. -0.5)

fIGURE C.S. The co-ordinates of the points P" P2 and p) with respect to the centre of asphere of unit radius represented by fl. r2 and fJ at times T,. T2 and T.,. This representsthe sphere routing in a clockwise direction by a quarter of a revolution per image.

z

x

FrGURE C.6. The four vectors rJ, ra, r', and r'z represent (with respect to the centreof the ball) two points on the surface of the balls at limes Tl and T2. TI and r'l make thesame angle with the spin axis. as do ra and r't. and this fact enables the calculation of thethree dimensional spin.

Using equations CS, C.9 and C.1O the direction cosines of the three points arc.

cosce I '" 0.707. COSj3I=0.5, cosY:'"' 0.5CO&o::: '"' 0.707, cosj3z'" 0.5. COSY2'" ·05

and cosO::) '" 0.707. COli{33"" -0.5. cos)')'''' -0.5.

158

From equations C.l I to C.14 L"C direction cosines of the spin axis are. cosce = I. C05(3 ".

o and cosy =: O. The magnitude of the spin can be calculated using equations 0.15 and0.16 as follows,

rose = I x 0.707 + 0 x 0.5 + 0 x 0.5 = 0.707e = 0.7854rad

. .:... of «0.707· .0707)' + W·5 ·0.5)' .• (0.5 .• 0.5)') _ 0707SIO Z'" rOI - 2Sln( .7854) - .

~rOl = 0.7854rad

and 4>"" = l.5708rad • 90"

Multiplying this by the direction cosines of the spin axis gives $er = 90", ¢>~=: (jJ and 4>)'=: 0°. If the time between images is o.(Xn seconds. say, then the magnitude or this spin is50 revolutions per second about the X-axis.

C.4. Limitations of the methodInitially, the spin calculation was not possible since there were only two images of theincoming ball in most of the photographs (Figure 2.10). A simple modification made itpossible to employ the above method using IWOpoints on two successive images of theball. The method is the same but four points arc now used (Figure C.6) and only thesubscripts change in equations C.S to C.16.

Image] (9PIC, (414,181)P,(415.190) C, P,P, (422, 190)

Direction of tIav/ I);rc:cuon of ro""u",

@ Image 2C, (33.7,9.1)p', (34.4, 9.9)p', (34.2, 8.6)

FIGURE C.7. A diagram showing the IWO dimensional co-ordinates of two points onconsccunve images of a golf ball from the photograph of an impacl The points representexactly the points seen on the photograph although the distance between images is not toscale.

involved subtracting the co-ordinates of the centre of the ball from the co-ordinates or thepoints. Since these were fairly near to each other. the combined error was large comparedto the result. In the calculation of the direction cosines for the spin axis there are manysubtractions. all increasing the error relative 10 the result. Sec lion 5 of this appendix

159

describes a BASIC program which calculates the three dimensional spin of a golf ballusing two points on two consecutive images. This program was used to analyse someimages ora golfbalJ impact, one of which is shown in diagrammatic form in Figure C.7.

Due to the large number of steps in the calculation of the spin it was not viable to calculateexactly an error for each result that was output. In order to get an estimate of the accuracyof the calculations the co-ordinates of the points were altered by the error in theirmeasurement, i.e. '! Imm. The co-ordinates were altered by :!: Imm so that allcombinations were accounted for and the effect on the calculation of the final spin noted.The co-ordinates of the centre of the first image were altered by :t lmm and the magnitudeof the total spin varied from 54.74 to I3S.8rads·l. Keeping the value of the centreconstant and varying the co-ordinates of the two points on image 1 in tum by ±Imm hadthe effect of altering the total spin from 55.65 to 165.29 rads-J. When the co-ordinateerror was reduced to ±O.5m.m the range of spin was 109.0 to 15 LOrads·1 and if it wasreduced still further to ±O.lmm the range was 132.50 to 138.46rads·1• The errors. in thecalculation of the final spin are too large to rely on the result when the error in each co-ordinate is ± lmrn. With an error of ±O.lnun. however, the range of values for the totalspin is small enough for the calculations to be considered accurate. This implies that theimages of the balls have to be considerably bigger and the scale a lot finer if thecalculations of three dimensional spin are to be accurate. It is therefore possible that thismethod would be more suitable for studying impacts of larger balls such as tennis. cricketor football.

C.S. A BASIC program to perform the three dimensional spin calculations

A large number of mathematical steps are required to perform the calculations above sothe mathematics were written into a BASIC program to increase the speed of the process;this is shown below. The language used was Microsoft BASIC for the AppleMacintoshPlus.

REl\t· .••• •.•.•.·······*1HREE DL'dENSIONAL SPIN ANALYSfS ••••.••.• • •• • ••••REM······· ..··••• ..•·•· ..•.······(3DSPIN.2P)··············· .••••••••••••••REM"'··························By S.J.Haake····························REM""fhis program calculates the spin of a sphere about its origin and gives the answerREM"as spin about the three dimensional axes. The program uses the two dimensionalREM··co-ordinates of two points at successive times from photographs ofbal! impacts.REM"The program initially alters the co-ordinates to allow for the relief displacementREM"(i.e the distortion of the image on the photograph due to the distance of the objectREM··away from the principal axis of the camera lens).

REM··Y 2, is the principle pointREM··YO(l).zD(I) is the centre of the first imageREM··Y(l.l),Z(l,1) is the first node on the first imageREM··Y(2,1),1..(2,1) is the second node on the first imageREM"The coordinates for the successive images are similar but with the second arrayREM··variable as 2,3 and 4.

REM··Dimension the arraysDIM Y0(4),ZO(4),Y(2,4),Z<2,4)

REM**Read in the dataFORJ%=l T04

READ YO(J%),ZO(J%)FOR 1%=1T02READ Y(I%J%),Z(I%J%)

NEXT)%NEXT}%READ Y,Z,F

160

REM**AdjUM the data to allow for the relief displacementFORI%=IT02

FORJ%d T04IF 4.S36901-(Y(I%,J%)-YO(J%»A2-(Z(I%~%)-ZO(J%»A2<0 THENX(I%,)%).O:GOTO SX(I%,J%).SQR(4.S36901-(Y(j%,J%)- YO(J%»A2-(Z(I%,J%)-ZO(J%»)A2)R(I%~%),SQR«Y(I%,J%)-Y)A2+(Z{I%,J%)-Z)A2)D(I%,J%).X(I%,J%)'R(I%,J%)'.0IS4YO(J%)=YO(J%)-Y:Y(I%~%).Y(I%~%)-YZO(J%).ZO(J%)-Z:Z{I%,J%).Z{I%~%)-ZK(I%,J%)= I-D(l%~%)IR(I%,J%)Y(I%,J%). Y(I%,J%)'K(l%,J%)Z(I%,J%).Z(I%,J%)'K(I%,J%)

Y(I%,J%).Y(I%,J%)-YO(J%):Z(I%,J%).Z{I%,J%)-ZO(J%)IF 4.S36901-Y(I%,J%)'2-Z(I%,J%)'2<0 THEN X(I%,J%).O:GOTO 6X(I%,)%).SQR(4.S36901- Y(I%,J%)A2-Z(I%,J%)'2)

6 NEXT J%NEXT 1%

REM .•..•Modiry cartesian coordinates to direction cosinesFORJ%.IT04

FOR 1%.1 '1'02X(I%,J%).X(I%,J%)I2.13Y(I%,J%). Y(I%,J%)I2.13Z{1%,J%).Z{I%,J%)/2.13NEXT 1%

NEXTJ%

REM*f<Calculate spin vector before impactIF Z{2,1)-Z{2,2). 0 THEN A.IE+20,GOTO 10A.(Z( 1,1)-Z{ 1,2»/(Z{2,1)- Z(2,2)l

10 IF A'(Y(2,1)-Y(2,2)-Y(I,I)+Y(I,2).OTHEN 20B.(X(1,I)-X(I,2)-A'(X(2,1)-X(2,2)))/(A'(Y(2,1)- Y(2,2»- Y( 1,1)+ Y( 1,2»C= I.B'B+«X(I,I )-X(1,2)+B'(Y( 1,1)- Y(I,2)))/(Z{I,2)-Z{I,I)))A2:GOTO 30

20 B.IE+20:C.IE+2030 Xb=SQR( IIC)

Yb=B'XbZb.SQR( 1-( 1+B'B)' Xb' Xb)

REM«·Calculate rotation before impactANGb.Xb'X( 1,1)+ Yb'Y( 1,1)+Zb'Z( 1,1)ANGb.-ATN(ANGbtSQR(-ANGb'ANGb+I»+1.570796R.4.26'SIN(ANGb)D.SQR«X( I, i)-X( 1,2»A'+(Y(I,I)_ Y(I,2))'2+(Z( 1,1)_Z(I,2»A2)ROTb=DIRROTb.2' ATN(ROTbtSQR(-ROTb'ROTb+ I»

REM"'CalcuialC spin vector after impactIF Z{2,3)-Z{2,4). OTHEN A=IE+20:GOTO 10A.!Z(I,3)-Z{I,4»/(Z(2,3)-Z{2,4»)

10 IF A'(Y(2,3)-Y(2,4»-Y(I,3)+Y(I,4).0 THEN 201l.(X( 1,3)-X( 1,4)-A '(X(2,3 )-X(2,4 »)l/(A '(Y(2,3)- Y(2,4»- Y( 1,3)+Y( 1,4»C.I·.B'B+«X(I,3)-X(I,4)+B'(Y(I,3)-Y(I,4)))/(Z{I,4)-Z{I,3)))A2:GOTO 30

20 B.IE+20:C·IE+2030 X•• SQR(I/C)

Ya=B*XaZa.SQR(I-{I+B'B)'X.'X.)

REM*·Calculalc rotation after impactANG.=X.' X( 1,3)+y.'Y( 1,3)+Za'Z( 1,3)

161

ANGa=-ATN(ANGalSQR( -ANGa' ANGa. 1».1.570796R=4.26+SIN(ANGa)D=SQ R«X( 1,3)-X( I ,4»A2+(Y( 1,3)-Y( 1,4))A2+(Z( 1,3)·Z( 1,4))'2)ROTb=DIRROTb·.2' AT!'iiROTbISQR( -ROTb'ROTb. I»

REMUOutput of dataPRINT "...• •• "Before impact····· ..PRINT "Spin Vector is ";:PRINT USING "+U~:';Xb;Yb;Zb

~~;I~X;~njsl~l~l~#~~~!;N~~;:+#~b~"~:~~Th'f.,b;:d)~R~~1~ ,:,!!:~p;?JfUSING "+N#U~";ROTb'Zb*F;:PRINT" radls."PRINT "Total rotation is ";:PRINT USING "+###.##";ROTb"'f;:PRINT" fad/sol

PRINT "· •.•. ··After impact·""''''''''''PRINT "Spin Vector is ";:PRINT USING "+IU##.";Xa;Ya;ZaPRINT "ROT,. ";:PRINT USING "+###.~#";ROTa*Xa+F;:PRINT" rad/s, ROTy• ";:PRINT USING "+N##.##"; ROTa*Ya*F;:PRINT" radis. ROTz = ";:PRINTUSING "+###.##";ROTa*Za*F;:PRINT" rad/s."PRINT "Total rotation is ";:PRINT USING "+##U#";ROTa*F;:PRINT" radJs"

ENDDATA 41.4,18.1DATA 41.5,19.0DATA 42.2,17.9DATA 33.7,9.1DATA 34.4,9.9DATA 34.2,8.6DATA 41.3,18.0OAT A 41.S,19.2DATA 42.2,17.9DATA 33.8,9.2DATA 34.4,9.9DATA 34.2,8.6DATA 31.0,145.250

162

Appendix D - Data on the impacts of two-piece and woundgolf balls

The following results were taken on a natural green ar Austerfleld Park. The PoaQn1Iua content of the green was 19%. the moisture content was 24%. the organicmatter content was 12.2% and the percentage of fines in the soil was 48%.

0.1 Two-piece golf balls

VELOCITY (ms- ANGLE(") SPIN rads-t) DEPTIIInitial mal initial mal mm ma mm)II ,.•ie.s

.1 I.5 '.U I

i I 1 .1 II :.3 I I. .1

:.3 II.I. :1I. 5 -I 4I. 5 ·11 .4I. 5 7

77 51.

15 .1.1

.1 .1

.1I '.4

57 -1 .1.7

7 4.7-·5

I. 1

4

TABLE D.l. A table showing the velocities. angles and spins before and after impactusing a two-piece golf ball.

163

0.2 'Vound golf balls

VELOCITY ms-t) ANGLE!") SPIN rads-! DEPTIlInJtI m trutr 111 initial lila mm)I II .1 4.I "~I -I 1.81 1 I .1I 45 4 17. I17 7 17. I .11. 4. 51.71. 1.0 .1

-II 7.0 .6-II 5

1. 1.15 .1

1. I I1. I

.11 "~

7. I .11

-I4 -I ,.~ .1

.1'.9.1

-4 1 .1.1

I -5 l.l .11

.14.

TABLE D.2. A table showing the velocities. angles and spins before and after impactusing a wound golf ball.

164

Appendix E • results of the photographs of impacts

TABLE E.l. A table containing the results from the 721 photographs of impacts. Thesubcsripts "i" and "f" denote the initial and final values respectively. Blank spaces occurwhere the variable was not measured.

'>;LOl Ar .>; ~I'JNms- de rees rads' mm

JK~>; I I I

A ERfiE 1 ,.~A 1 '.SAU TERFI L 1'.9 4A rs EL 1 .\ 5 1.8., ER 1 o 1 1.5A E 1 5 1 .1 I .JA " .4 4 1 .1A I 5 1A ~.. I. .6A .1A I. - -IA 1A I.A I.A 4 ,1.

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I I i.' .1A 1 4 1A I .< 1'.9 1A I.A I.A I.A -IW.4

A -!Z9.6 1.1A I. .1

165

AN iLt: SPIN Ums- de rees rads- mm

A T R lEL 5. 4 5 I.A -4 I. 1A TA T' 7. 1 l.gA 1 '.1A 4 1 ;.6AA 4 5 -IA -I .1A .1 7.6A 45 5 1.9A FE .1 .1A .4 7 1 .1A .1 1 .5 .1A 7. 1 -5 1.1 .1A 7 8.9 1 .7A .1A .1B .1

I.4 ·111.7 J.9 .4

B -I 71.gB -1

11.1 I 4 1 8.4E o.

IU.11. .111. .1 .1

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Bl.".1 1 1

1.1 .1B -11. 1B -128.4 I. I.

R I. 1B ..FR 4 i:

E Y .1 I. 1.17

I. .1 ,.0 z.:." 1.8

I.-1

.1 .1 r.z1 1.9

I ,.3 .1 .6..6

166

ms·· de rees rods' mm,.6 0.1 I ·111. ·.S

I JII. 1.6

I 11. 1.1B I I I.

B I 7 I .- I.

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A I 4<> -\1 •.• .iHA I 40 -11 1.9 ! 4.

A I 40 1 ·11 .1.9W 1 40 1.9 .r

A I 47 7 .7 I.

A 1 40 ..4. ., 540 0 .t

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171

" uLI<de rees., rads-rns- mm

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172

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)NORI;K. I. V'NURICK.'NURICK.'NORICK. I. 4INORICK -I

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173

v ,LUl AN u; SPIN Ums- de rees rads- mm

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