TP A.14 The effects of cut angle, speed, and spin on object ball throw

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1 echnical proof echnical proof TP A.14 The effecs of cu angle, speed, and spin on objec ball hrow supporing: The Illusraed Principles of Pool and illiards hp://billiards.colosae.edu by Daid G. Alciaore, PhD, PE ("Dr. Dae") originally posed: 7/1/ las reision: 9//17 See TPs A., A.6 and A.8 for background inforaion and illusraions. For a cue ball wih boh erical-plane (draw or follow) and horional-plane (side) English, he elociy of he poin of conac () beween he cue ball and he objec ball, a ipac is: ) cos sin ( ) ( / j i k i j r x O (1) So k j i k j i x y x cos sin cos () Expressing his ecor in angenial and noral coponens (see TP A.) gies: k n k n x n cos cos sin (3) Therefore, he relaie "sliding" elociy ecor for he poin of conac beween he cue ball and objec ball can be expressed as: k rel (4) because he noral (n) coponen, which creaes he ipac forces, does no conribue o he relaie sliding. The fricion force on he objec ball acs in he direcion of he sliding elociy ecor: k e e k k e rel rel ()

2 Fro Equaion 8 in TP A., he noral ipulse beween he cue ball and objec ball, assuing a perfecly elasic collision, is: F n cos (6) Fro TP A.6, and using Equaions 3 and, he axiu possible fricion ipulse coponen ha conribues o objec ball hrow, based on he liiaion of fricion, is: F ax F e n cos sin sin cos x (7) Anoher lii on he axiu possible fricion ipulse is based on he requireen ha he relaie sliding oion beween he cue ball (C) and objec ball (O) canno reerse direcion during ipac. The relaie angenial speed beween he balls afer ipac is gien by: rel (8) C C where he ers on he righ side indicae pos-ipac speed coponens in he angenial direcion and he ers indicae pos-ipac roaional speeds. O O In he liiing case, rel will go o ero during ipac (i.e., sliding ceases during ipac). In his case, he axiu pos-ipac angenial speed he objec ball can hae is: O ax C C Fro TP A. and ipulse-oenu principles, he ers in Equaion 9 can be expressed as: C O F O F ax C sin F F ax ax ax (1) ax O F ax (9) (11) (1) (13) Subsiuing Equaions 1 hrough 13 ino Equaion 9 yields he axiu possible fricion ipulse coponen ha conribues o objec ball hrow, based on he relaie speed kineaics consrain: F ax sin 7 (14)

3 Using Equaions 7 and 14, we can now deerine he pos-ipac objec ball angenial speed based on which effec is he os liiing (fricion or kineaics): F cos sin cos sin O in x 1 7 (1) The noral coponen of he pos-ipac objec ball elociy, using Equaion 6, is: O n Fn cos (16) Now he objec ball hrow angle can be calculaed wih: 1 O O an (17) O n Typical alues for he paraeers used in he equaions: μ.6 aerage coefficien of fricion beween he balls 1.1in ball radius conered o eers.9 3ph aerage cue ball speed conered o eers/sec s ω roll naural-roll spin rae Typical speeds (conered o eers/sec): slow 1ph slow.447 s ediu 3ph ediu s fas 7ph s fas 3.19

4 Model of fricion based on Marlow daa (Table 1 on p. 4 in "The Physics of Pocke illiards," 199) d.1sin( 4) μd d 1sin( 4) μd.6 d 1sin( 4) μd Noe - he Marlow speed daa is uliplied by sin(4 ) o reflec he relaie angenial speed coponen for he sho in his experien (which had a 4 cu angle). The fricion s. speed relaion sees o follow (or le's assue i follows) a relaion of he for: u ( ) = a be c We can sole for he coefficiens fro he se of daa aboe using: iniial guesses: a μd b μd μd c 1 ln d μd a b a.1 b.1 c.98 Gien μd a b e c d 1 = 1 μd a b e c d = μd a b e c d 3 = 3 a b c Find( ab c ) a b c μ( ) a be c Plo of experienal fricion daa wih heoreical cure fi i d 3.1 μd i.1 μ( ). 1 1 d i

5 MahCAD forulaions of Equaions 1 hrough 17, using he aboe forulaion for fricion: sin( ) ω ω x cos ( ) rel ω x ω aan ω x ω if = rel ω x ω in μ rel ω x ω cos( ) 1 rel ω x ω 7 cos( ) sin( ) ω These wo equaions describe all hrow effecs (see he plos below)! collision-induced hrow s. cu angle for arious-speed naural-roll shos: 1 9 slow slow ediu ediu fas fas OTTOM LINE: slower speed resuls in ore collision-induced hrow; and he aoun of hrow increases wih cu angle, bu i leels off a higher cu angles.

6 collision-induced hrow s. cu angle for sun shos a arious speeds: slow ediu fas These plos can be copared o ob Jewe's experienal daa locaed a: hp:// and wih he daa presened in y Sepeber '6 insrucional aricle. The heoreical plos and experiens agree ery closely in boh cure shape and hrow alues. OTTOM LINE: The aoun of hrow is independen of speed a sall cu angles. Throw is larges in he half-ball hi range (3-ree cu angle range). A larger cu angles, hrow is larger for slower speeds.

7 collision-induced hrow s. cu angle for shos wih arious aouns of erical plane spin (follow or draw), copared o sun 4 ω roll.ω roll.ω roll ( ) OTTOM LINE: collision-induced hrow is greaer for sun shos close o a 1/-ball hi (3 ree cu angle).

8 spin-induced hrow for sraigh-on shos (-ree cu angle) wih arious aouns of forward roll (or draw) and sidespin ω 1.ω roll 1.ω roll 1.ω roll (see TP A.1) 4 ω roll ω.ω roll ω.ω roll ω ω ω These plos are siilar o hose in Figure 4.4 on p. 4 of on Shepard's "Aaeur Physics for he Aaeur Pool Player," 3rd ediion, Howeer, here, a ore accurae odel of collision dynaics and kineaics inoling fricion is used. OTTOM LINE: spin-induced hrow is greaes, and os sensiie o sidespin, wih sun shos and for ediu aouns of spin.

9 spin-induced hrow for sraigh-on sun shos (-ree cu angle) a arious speeds, wih differen English percenages pe 1% 9% 1% (see TP A.1 and TP A.) side spin = SF (/) SF = 1. (pe) 1 slow slow 1.pE ediu ediu 1.pE fas fas 1.pE 1 1 pe % OTTOM LINE: spin-induced hrow is independen of speed for sall aouns of English and larges a abou % English for a slow sho. Afer a poin, ore English doesn' creae ore hrow.

10 cobinaion of collision- and spin-induced hrow for a half-ball hi (3-ree cu angle) wih arious aouns of forward roll (or draw) and sidespin 3 ω 1.ω roll 1.ω roll 1.ω roll (see TP A.1) 4 ω roll ω.ω roll ω.ω roll ω ω ω OTTOM LINE: For a half-ball hi, hrow is greaes for a sun sho wih no sidespin or wih ouside English wih a spin rae facor of 1 (see TP A.1). For ouside English wih a spin rae facor of., here is no hrow.

11 cobinaion of collision- and spin-induced hrow for a half-ball hi (3-ree cu angle) sun shos a differen speeds and wih arious aouns of sidespin fas 3 ω roll ω 1.ω roll 1.4ω roll 1.ω roll (see TP A.1) 1 slow ω ediu ω fas ω ω ω ω slow ediu fas OTTOM LINE: For a half-ball hi sun sho, hrow is greaes wih lile or no sidespin or wih significan ouside English. The aoun of hrow is os sensiie o he aoun of sidespin near he hrow-cancelling ouside English poin.

12 hrow s. cu angle for arious ypes of ypical side-english, slow speed, sun shos: 1 9 slow ω roll ω sun ω none ω inside ω roll ω ouside ω roll ω gearing ( ) sin( ) ω sun ω none ω sun ω inside ω sun ω ouside ω sun ω gearing ( ) These plos agree wih he qualiaie plos in Figure 6-1 on p. 74 of Jack Koehler's "The Science of Pocke illiards," OTTOM LINE: Inside English increases hrow a sall cu angles. Ouside English reerses collision-induced hrow, and has axiu effec a sall cu angles. "Gearing" ouside English resuls in absoluely no hrow.

13 hrow s. cu angle for arious ypes of % side-english, slow speed, sun shos: slow ω roll ω sun SF 4 % ω none ω inside SF ω roll ω ouside SF ω roll ω gearing ( ) sin( ) ω sun ω none ω sun ω inside ω sun ω ouside ω roll ω gearing ( ) 4 6 hrow s. cu angle for arious ypes of % side-english, ediu speed, follow/draw shos: fas ω roll ω sun SF 4 % ω none ω inside SF ω roll ω ouside SF ω roll ω gearing ( ) sin( ) ω roll ω none ω roll ω inside ω roll ω ouside ω roll ω gearing ( ) 4 6

14 hrow s. cu angle for arious ypes of % side-english, slow speed, sun shos: slow ω roll ω sun SF 4 % ω none ω inside SF ω roll ω ouside SF ω roll ω gearing ( ) sin( ) ω sun ω none ω sun ω inside ω sun ω ouside ω roll ω gearing ( ) 4 6 hrow s. cu angle for arious ypes of % side-english, ediu speed, follow/draw shos: ediu ω roll ω sun SF 4 % ω none ω inside SF ω roll ω ouside SF ω roll ω gearing ( ) sin( ) ω roll ω none ω roll ω inside ω roll ω ouside ω roll ω gearing ( ) 4 6

15 hrow s. cu angle for arious ypes of 1% side-english, slow speed, sun shos: slow ω roll ω sun SF 4 1% ω none ω inside SF ω roll ω ouside SF ω roll ω gearing ( ) sin( ) ω sun ω none ω sun ω inside ω sun ω ouside ω roll ω gearing ( ) 4 6 hrow s. cu angle for arious ypes of 1% side-english, ediu speed, follow/draw shos: ediu ω roll ω sun SF 4 1% ω none ω inside SF ω roll ω ouside SF ω roll ω gearing ( ) sin( ) ω roll ω none ω roll ω inside ω roll ω ouside ω roll ω gearing ( ) 4 6

16 Effecs of cling/skid/kick a differen cu angles: aan ω x ω μ μ : fricion uliplier (1: noral, >1: diry condiions, up o 3: cling/skid/kick) if = rel ω x ω in μ μ rel ω x ω rel ω x ω cos( ) cos( ) 1 7 sin( ) ω Throw and cling/skid/kick is axiu for a slow, sun sho: slow ω x ω μ _noral 1. μ _diry 1. μ _cling. ω x ω μ _noral ω x ω μ _diry ω x ω μ _cling For sun shos, he axiu aoun of hrow is norally a close o a 1/-ball hi (3 ree cu angle); bu wih clingy condiions, axiu hrow occurs a larger cu angles (-6 rees). For sall cu angles (less han abou 3 rees), he aoun of hrow is he sae regardless of he aoun of fricion.

17 For a slow rolling-c sho, here's how cling/skid/kick affecs shos: ω x ω x ω μ _noral ω x ω μ _diry ω x ω μ _cling For rolling-c shos, hrow increases wih cling a all cu angles, and is larger a larger cu angles.

TP B.2 Rolling resistance, spin resistance, and "ball turn"

TP B.2 Rolling resistance, spin resistance, and ball turn echnical proof TP B. olling reiance, pin reiance, and "ball urn" upporing: The Illuraed Principle of Pool and Billiard hp://billiard.coloae.edu by Daid G. Alciaore, PhD, PE ("Dr. Dae") echnical proof originally