Learning from a Golf Ball

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1 Session 1566 Learning from a Golf Ball Alireza Mohammadzadeh Padnos School of Engineering Grand Valley Sae Uniersiy Oeriew Projecile moion of objecs, in he absence of air fricion, is sdied in dynamics classes and exbooks 1,, 3. In his sdy sdens learn abo he effec of air drag on he projecile s heigh and range. In he process of achieing an ndersanding of he effec of air fricion on projecile moions, sdens, in dynamics class, learn how o ilize programming feares of Mah-Cad sofware o sole a sysem of non-linear, firs-order, ime dependen, ordinary differenial eqaions. Sdens also learn abo he applicaion of difference eqaions and Rnge-Ka mehod o obain a solion for a sysem of nonlinear, firs-order, ime dependen, ordinary differenial eqaions. Physical ndersanding of effecs of air resisance on a golf ball s rajecory is achieed by comparing he resls of hese differen approaches wih he resl of moion in he absence of he air drag. Formlaion The drag force de o air fricion on a golf ball can be esimaed by: 1 CAV 1 Where is he drag force on he ball, V is he speed of he ball, is he densiy of he air, A is he projeced area of he ball normal o he air flow, and C is he drag coefficien. Neglecing he srface roghness of he ball, he drag coefficien C depends on, air iscosiy, air densiy, ball speed, and ball diameer. Tha is: C f,, V, d Where is he air iscosiy, is he air densiy, V is he speed of he ball, and d is he diameer of he ball. The effec of all hese parameers on he drag coefficien can be lmped ino a single dimensionless parameer known as ynolds nmber. Tha is: Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

2 C = f 3 Vd Where 4 The drag force, being of fricional nare, acs in he opposie direcion of he elociy of he golf ball. Applicaion of Newon s second law o he projecile moion of he ball renders he goerning eqaions of moion. One can wrie: V y mg j i x F.B.. F ex ma 5 Obsering he free body diagram aboe, eqaion 5 renders: mg ma 6 Where m is he mass of he ball, g is he graiaional acceleraion, dv a 7 d a is he acceleraion of he ball, and V is he ball elociy. efining he ni ecor V, one cold obain: V V 8 V Sbsiing eqaion 1 ino eqaion 8 and he ocome of ha ino eqaion 6, one obains: mgj 1 C AVV ma Sbsiing 7 ino 9 and noicing ha he cross-secion of he ball is eqaion 9 becomes: 9 A 4 d, Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

3 dv mgj CVd V m 1 8 d Wriing he elociy of he ball in erms of is x and y componens V i j and afer some algebraic maniplaion eqaion 1 becomes: d i j C gj i j d 8 m d Considering ha = dx/d, and = dy/d, where x is he horizonal disance and y is he m erical disance raeled by he ball, and noicing ha he erm in eqaion 11 d is a consan and has dimension of ime, one can arrie a he goerning eqaions of moion pon wriing 11 in is x and y componens: dx d dy d 1 d C d 8 d C g d 8 Eqaion 1 sbjec o iniial condiions below: 11 x y V V cos sin 13 Make he sysem of non-linear, ime dependen, ordinary differenial eqaions, which describes he fligh of he golf ball. In eqaion 13 V is he lanching speed of he golf ball and is he lanching angle of he ball wih respec o horizon. The drag coefficien in he differenial eqaions of moion is calclaed sing a correlaion eqaion for a smooh sphere C Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

4 The aboe correlaion deiaes from he sandard drag cre for spheres by 4% o 6% for ynolds nmbers p o A he lower end of he ynolds specrm he aboe 4 correlaion renders drag coefficien of a sphere in Soke s flow, ha is C. The aboe correlaion will no incorporae he effec of srface roghness. Srface roghness is o indce rblence, which in rn redces he pressre drag. To simlae he effec of dimples on he golf ball, we calclae he drag coefficien by considering ynolds nmber a 31 5 and sing he following eqaion C To show he significan role, which dimples play on he range and heigh of he golf ball projecile, we compare he resls of rogh srface calclaions wih hose of smooh srface balls. Solion A closed form solion o he sysem of non-linear, ime-dependen, ordinary differenial eqaions 1 sbjec o iniial condiions 13 is no possible. Three differen nmerical approaches are employed in his sdy o calclae he pah of a golf ball; namely, difference eqaions scheme, Rnge-Ka mehod, and a Mah-Cad bil-in fncion. This is he firs ime ha he sdens ry o se nmerical mehods o sole sysem of differenial eqaions in he crriclm. Pedagogically, i is ery sefl if sdens ry differen approaches and compare he resls of hese differen approaches, for a pracical applicaion sch as he golf ball moion. To simlae he moion in Mah-Cad we consider he fligh of a ball of m = 43 g, d = 45 mm in air of = N.s/m, = 1.5 kg/m 3. The lanching speed and lanching angle of he golf ball are 36 m/s and 3 o wih horizon respeciely. Approach I- ifference Eqaions Scheme In his approach we approximae he ariables and heir deriaies in eqaions 13 as follows: dx x x d dy y y d d d d d Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion 16 Page

5 Sbsiing eqaions 15 ino he goerning eqaions of moion, one arries a he difference eqaions of moion as: y y x x 17 g C C 8 8 Soling he aboe sysem of eqaions for he ariables a ime, one obains: y y x x C g C C C 18 The aboe sysem of eqaion is hen soled by marching hrogh ime from iniial =, when he ball was lanched, nil he ime when he ball his he grond. A ime sep of =.1 sec is adoped in his simlaion. Appendix A represen he Mah-Cad program sed o implemen his approach. Approach II- Forh-Order Rnge-Ka Mehod A forh-order Rnge-Ka 5 solion echniqe is employed o sole he sysem of nonlinear ime-dependen firs-order differenial eqaions. Appendix B shows he deail Mah-Cad program o perform his inegraion. Approach III- Mah-Cad Bil-in Fncion rkfixed Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

6 The Mah-Cad bil-in fncion rkfixed 6, which is an implemenaion of forh-order Rnge-Ka mehod, is sed o sole he sysem of non-linear differenial eqaions 1. The deail of calling his fncion in Mah-Cad is also shown in Appendix B. sls and iscssions Figre I, aken from Appendix A, shows he projecile moion of he golf ball for boh smooh and rogh srface cases. The simlaion resls for smooh ball and rogh ball rajecories were achieed by soling he difference eqaions 18 of he goerning differenial eqaions of moion. The rogh ball case resls in a heigh of rajecory of 15.5 m and a range of 11.7 m. The corresponding ales for he smooh ball in Figre I are m and m respeciely. I is worh noing ha he rajecories for boh cases are idenical p o almos 8% of he smooh ball fligh. The high ynolds nmber a he sar of he projecile moion of he smooh ball renders a ery low drag coefficien, which resls in a rajecory of he ball similar o ha of a rogh ball. Noice also ha he effec of air resisance is more prononced on he heigh of he golf ball han on is range. The reason being ha he ball deceleraes more in he erical direcion han in he horizonal direcion de o graiy. Figre II, aken from Appendix B, depics he resls of he golf ball simlaion for smooh and rogh ball srfaces, sing approaches II and III o sole he goerning differenial eqaions of moion. Figre II also shows he resl of he drag free simlaion. Comparison was made beween he Rnge-Ka approach and Mah-Cad rkfixed approach for smooh ball srfaces. The resls were idenical as expeced. Incidenally, we ge he same idenical resls when we compare he resls of 3 differen approaches for smooh ball simlaion. Tha is o say ha he 3 differen solion sraegies yield a range of m and a heigh of m for he smooh srface golf ball rajecory. Figre II indicaes ha rogh balls hae a rajecory mch closer o he drag free one. The reason being ha he rogh srface indces rblence which in rn sends he flow separaion poins frher down-sream of he flow on he srface of he ball and redces drag coefficien. Heigh s Horizonal Trael of he Ball Heigh m N 4 M N Horizonal Trael m Figre I- Comparison of Approach I sl Wih he rag Free Ball Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

7 Heigh m Z M yx Q 1 Heigh s Horizonal Trael of he Ball 5 1 x Horizonal Trael m 1 Figre II-Comparison of Approach II and III sls Wih Each Oher for Smooh Ball case, and Wih he sls of Rogh and rag Free Balls The nmerical ales of he range and he heigh of he projecile in case of he rogh srface golf ball were 11.7 m, and 15.5 m respeciely. The corresponding ales for he drag free fligh are m, and m respeciely. This made i clear o he sdens as o why here are dimples on he srface of he golf balls. The Table below smmarizes he resls of all he simlaed cases. Projecile ifference Eqaions sls for Smooh Ball Heigh m Range m Conclsion Comparison of sls of he Golf Ball Projecile Simlaion Rnge-Ka sls for Smooh Ball Rkfixed sls for Smooh Ball Rnge-Ka sls for Rogh Ball rag Free sls Table I Golf ball rajecories for smooh and rogh balls in he presence of air resisance were ealaed by employing difference eqaions schemes, Rnge-Ka mehod, and Mah- Cad rkfixed fncion o sole he goerning differenial eqaions of moion. I was realized ha resls from he 3 differen solion echniqes were he same See Table I. I was also obsered ha by sing a rogh srface ball insead of a smooh one, one cold achiee a 14% improemen in he range of he golf ball and a 31.5% improemen in he heigh of he ball See Table I. Sdens learned abo he imporan effec ha dimples on he srface of he golf ball hae on he ball s rajecory. Sdens learned how o sole a sysem of non-linear firs-order ime dependen differenial eqaions in 3 differen Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

8 ways. They also learned abo he programming feares of Mah-Cad sofware. Oerall he projec was a sccess and sdens feedback was ha hey learned sbsanially from his projec. Bibliography 1. Beer, F. P., and J. E. Rssell: Vecor Mechanics for Engineers- ynamics, 6 h. Ediion, McGraw Hill, Meriam, J.L., and L.G. Kraige: Engineering Mechanics- ynamics, 4 h. Ediion, John Wiley, Riley, W.F., and L.. Srges: Engineering Mechanics- ynamics, John Wiley, Clif, R., and W.H. Gain: The moion of paricles in Trblen Gas Sreams, Proc. Chemeca 7, Vol. 1, p 14-8, Ayyb, B.M., and R.H. McCen: Nmerical Mehods for Engineers, Prenice Hall, Mah-Cad 7 User s Gide, MahSof Inc., ALI R. MOHAMMAZAEH is crrenly assisan professor of engineering a Padnos School of Engineering a Grand Valley Sae Uniersiy. He receied his B.S. in Mechanical Engineering from Sharif Uniersiy of Technology And his M.S. and Ph.. boh in Mechanical Engineering from he Uniersiy of Michigan a Ann Arbor. His research area of ineres is flid-srcre ineracion. Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

9 APPENIX A ifference Eqaion Solion of he Projecile Moion of he Ball for Boh Smooh and Rogh Golf Ball Cases Ball iameer and Air Properies d.45 in N sec m 1.5 kg m 3 rag Coefficien for Smooh Ball J = 1 C d.687 d d rag Coefficien for he Rogh Ball J = I is Assmed ha he Roghness of he Srface is Simlaed by incorporaing a High Vale of ynolds Nmber ino a Correlaion, e o Roghness Indced C Rogh The Programming par, Which Implemens he solion of Sysem of Nonlinear, Time-Independen Firs Order ifferenial Eqaions of he Golf Ball Trajecory By ifference Approach Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

10 m.43 f V j g 9.81 i 18 m d i Vcos i Vsin x i y i i.1 s i while i i d C C i i if j 1 C C Rogh if j i i 1 i i x i y i y i i1 i1 x i1 y i1 i s i C 16 i1 C 1 16 C 16 i1 C 1 16 i1 i i1 i H agmen x y H g Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

11 Calling Up he ifference Fncion for he Smooh Ball Case J = 1 M f Heigh s Horizonal Trael of he Ball Heigh m M M 3 Horizonal Trael m Calling Up he ifference Fncion for he Rogh Ball Case J = N f 3 36 Comparison of he Heigh and Horizonal Trae of he Golf Ball for Smooh and Rogh Srface Cases ashed Ble - Smooh Srface d - Rogh Srface In he Graph Below Heigh s Horizonal Trael of he Ball Heigh m N 4 M N 3 Horizonal Trael m Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

12 APPENIX B 4h. order Rnge_Kaa Sysem of ifferenial Eqaions Program RK4_sys f hxiniial n x xiniial for i i n hi for i n 1 k1 hf i x i h k h f i x i k1 h k3 h f i x i k k4 hf i hx i k3 x i1 x i s agmen x T s k1 k k k4 6 Applicaion of RK4_sys o Projecile Problem Inp aa m.43 d.45 h.1 kg m sec g Time Consan: m s Nsec m kg m 3 V 36 3 m sec de n 18 m d Golf Ball Srface Indicaor J = 1, for Smooh Srface, J =, for Rogh Srface j 1 Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

13 Iniial Condiions xiniial V cos 18 V sin 18 rag Coefficien fncion for Smooh Srface J = 1 C d.687 d d 1.16 rag Coefficien for Rogh Srface J C Rogh The eriaie Fncions Correlaion o calclae rag coefficien is sed in if Saemen fx if j 1C x x 3 if j 1 C x x 3 x x 3 C Rogh 8 C Rogh 8 x x 3 x x 3 d d x 3 g x Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

14 Calling Up The RK4 Program M RK4_sys f h xiniial n Projecile Moion in he Absence of rag x M 1 gx yx an.54 x 36cos.54 Heigh s Horizonal Trael of he Ball Verical Trael m M yx M 1 Horizonal Trael m Using Bil in fncion in Mahcad rkfixed To Sole Sysem of nonlinear O..E. x 36 cos sin 3 18 Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

15 x if j 1C x x 3 if j 1 C x x 3 x x 3 C Rogh 8 C Rogh 8 x x 3 x x 3 d d x 3 g x Calling Up he rkfixed Fncion Z rkfixed x 1 1 rkfixed sl Projecile Heigh s Horizonal Trael 4 Heigh m Z 5 1 Z 1 Horizonal Trael m Simlaion of Rogh Ball J = Using RK4 Fncion Z 1 M 1 x Z 1 j Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

16 fx if j 1C x x 3 if j 1 C x x 3 x x 3 C Rogh 8 C Rogh 8 Q RK4_sys f h xiniial n x x 3 x x 3 d d x 3 g x x Q 1 Comparison of he sls of Rogh and Smooh Balls,Using RK4 and rkfixed Fncions, wih rag Free Case Heigh s Horizonal Trael of he Ball Heigh m Z M yx Q x Horizonal Trael m Proceedings of he 4 American Sociey for Engineering Edcaion Annal Conference &Exposiion Copyrigh 4. American Sociey for Engineering Edcaion Page

1. The graph below shows the variation with time t of the acceleration a of an object from t = 0 to t = T. a

1. The graph below shows the variation with time t of the acceleration a of an object from t = 0 to t = T. a Kinemaics Paper 1 1. The graph below shows he ariaion wih ime of he acceleraion a of an objec from = o = T. a T The shaded area under he graph represens change in A. displacemen. B. elociy. C. momenum.