technical poof TP A.4 Pot-impact cue ball tajectoy fo any cut anle, peed, and pin uppotin: The Illutated Pinciple of Pool and Billiad http://billiad.colotate.edu by Daid G. Alciatoe, PhD, PE ("D. Dae") technical poof oiinally poted: 1/4/05 lat eiion: /3/08 initial cue ball diection θ c final cue ball diection cue ball tajectoy tanent line object ball y x 90 deee aimin line impact line initial cue ball elocity (pe impact) cut anle (
The tanlational equation of motion fo the cue ball (afte impact) i: whee and F F ma m & (1) i the fiction foce between the table cloth and the cue ball duin the cued tajectoy i the elocity of the cente of the cue ball. The otational equation of motion fo the cue ball (afte impact) i: whee and ω M α M I mr & ω 5 i the moment of the fiction foce about the cente of the cue ball i the anula elocity of the cue ball. () The elocity of the contact point () between the cue ball and the table cloth i: ω R ( iˆ ˆ) j ( ω iˆ ω ˆ) j ( Rkˆ) ( Rω )ˆ i ( Rω ) ˆj x y x y x y y x (3) Note that z-axi pin (ω z, eultin fom ide Enlih) would hae no effect on the contact point elocity, and theefoe would not affect the emainde of thi analyi. The fiction foce (µm) oppoe the lip, in a diection oppoite fom the elatie lip elocity diection: F m µ µ mˆ (4) The fiction moment can be expeed a: M R F µ mr Rˆ ˆ ( ) ( 5) Takin the time deiatie of the left ide of Equation 3, we can elate the linea and anula acceleation: & & & ω R (6) Subtitutin Equation 4 into Equation 1 ie the cue ball acceleation: & µ ˆ () Subtitutin Equation 5 into Equation ie the cue ball anula acceleation: & 5µ ω ( Rˆ ˆ ) R (8)
Theefoe the lat tem in Equation 6 can be witten a: R 5 µ & 5 ω ( Rˆ ˆ ) R µ ˆ R ( 9) Subtitutin Equation and 9 into Equation 6 ie: & 5µ µ µ ˆ ˆ ˆ (10) The followin concluion can be made concenin Equation 10: The elatie elocity ecto, and theefoe the fiction foce ecto (ee Equation 4), doe not chane diection!!! The elatie lip peed low to 0 and emain 0 theeafte (i.e., the cue ball tat ollin without lippin at a cetain point and continue to oll in a taiht line). Alo, fom Equation 4 and, the fiction foce ecto and the cue ball acceleation ae contant (in manitude and diection). Theefoe, the cue ball tajectoy will be paabolic, jut a with any contant acceleation motion (e.., pojectile motion). Fom Equation 10, it i clea that the elatie peed chane accodin to: Theefoe, the elatie peed aie accodin to: µ & (11) whee o µ ( t) o t (1) i the initial elatie peed (immediately afte object ball impact). So, now, the elatie elocity ecto i known oe time: ( t) ( o µ t) ˆ o (13) If the initial cue ball linea and anula elocitie (immediately afte object ball impact) ae pecified, Equation 3 ie: ( 0) (0) ω(0) R ( Rω )ˆ i ( Rω ) ˆj iˆ ˆj ox oy o ˆ o (14) The initial elatie elocity manitude (peed) i ien by: o ox oy ( Rω ) ( Rω ) (15)
and the diection of the initial elatie elocity (which emain contant) i: ˆ o ( ox iˆ o oy ˆ) j ( Rω ) ( Rω ) iˆ o o ˆj (16) Uin Equation 16 in Equation, we now know the cue ball acceleation & µ ˆ o (1) The olution to thi equation i: (18) ( t) (0) µ tˆ o Thi equation applie only while the cue ball i lidin. When lidin ceae, the cue ball moe in a taiht line tanent to the tajectoy at that point. The time it take fo lidin to ceae can be found fom Equation 1: µ ( t) o t 0 So the cue ball path will be cued only fo the followin duation (afte object ball impact) o t µ (19) (0) The final deflected anle of the cue ball path can be found by lookin at the lope of the tajectoy at the time ien by Equation 0. Fom Equation 16 and 18, uin Equation 0, the final component of the cue ball elocity ae: and xf yf µ ( Rω ) 1 x ( t) t ( Rω ) o o ( 5 Rω ) µ ( Rω ) 1 y ( t) t ( Rω ) ( 5 Rω ) (1) () Theefoe, the final deflected cue ball anle i: θ c tan 1 xf yf tan 1 5 5 Rω Rω (3) The final ball elocity f (Equation 1 and ) can alo be expeed in the followin ecto fom: f 5 o ωo (4) whee o i the initial pot-impact elocity, and ω o i the initial anula elocity. Inteetinly, the final elocity doe not depend on fiction µ. 5/ (1.4%) of the final elocity come fom the initial elocity ecto ( o ) in the tanent-line diection, and / (8.6%) come fom the initial pin elocity ecto (ω o x ). The ecto i taiht up fom the etin point to the cente of the ball (i.e., -R, elatin it to the R ecto aboe).
Equation 18 can be inteated to find the x and y coodinate of the cue ball tajectoy: µ x( t) t ox t (5) o µ y( t) t oy t (6) o Now it i clea that the tajectoy i a paabola (ee alo the paaaph afte Equation 10). Equation 5 and 6 apply only fo the time peiod ien by Equation 0. Gien the initial cue ball peed () in the y diection, and nelectin the fiction between the cue ball and object ball (fo now), the pot-impact cue ball peed and peed component ae (ee the fiue below and TP 3.1 fo moe detail): o in( in( co( () (8) in ( (9) co( φ o in( φ in ( in( co( If we aume the cue ball ha no y-axi pin (i.e., ω 0, which mean the hot ha only follow, daw, o tun), then Equation 15 (uin Equation 8 and 9) become: o Rω ( Rω) in( co ( in( in( (30) whee ω i the initial pin of the cue ball about the x axi. And fom Equation 14, in( co( (31) ox Rω in ( Rω (3) oy
And fom Equation 3, θ c 1 5 1 5 in( co( tan tan 5 Rω 5 in ( Rω (33) Uin Equation 8 thouh 3 in Equation 5 and 6, the cue ball tajectoy become: x( t) t in( co( µ t co( Rω co ( in( in( (34) y( t) t in ( Rω µ in( t in( Rω co ( in( in( (35) Fom Equation 0 and 30, we ee that Equation 34 and 35 apply only fom time 0 to time: in( t µ Rω co ( in( in( (36) If the cue ball i ollin without lippin at object ball impact, then: and Equation 33 educe to: ω R (3) 1 in( co( θ tan (38) c in ( 5 Thi aee with the famou eult fom the 198 Wallace and Schoede pape, which fomed the bai fo the 30 deee ule (ee TP 3.3). Alo, fo a ollin cue ball, Equation 4 (uin Equation and 3) become: f 5 5 o ωo in φ tˆ ˆj (39) whee t i the tanent line diection and j i the oiinal diection of the ollin cue ball (i.e., the aimin line). The fiue below how the implication of thi ey ueful eult, uin imila tianle. The final ball diection i at / of the ditance x between the tanent line and aimin line, meaued pependicula to the tanent line (i.e., paallel to the impact line). Thi eult i tue fo a ollin cue ball at any cut anle and peed.
j: cue ball aimin line 5 x f : final cue ball diection 5 φ x t: tanent line o 5 o in( o In Bob Jewett' July '08 Billiad Diet aticle, he how how u can ue the cue tick to help pedict the final cue ball diection fo a ollin cue ball hot. If u hold the cue tick (of lenth "x") pependicula to the tanent line (i.e., paallel to the impact line), with one end of the cue tick on the aimin line and the othe end on the tanent line, then the final diection of the cue ball will be at the / point alon the cue (at / x) fom the tanent line. Now, een thouh hot peed doen't affect the final diection of the cue ball, it doe affect the path to the final diection, o thi alo need to be taken into conideation when pedictin whee the cue ball will tael (ee the plot below and my June '05 Billiad Diet aticle).
Now we will look at ball tajectoie fo aiou type of hot. Equation 34 and 35 decibe the eneal tajectoie fo follow, daw, and tun hot. They apply only duin the time inteal ien by Equation 36. Hee ae the paamete ued in the equation alon with MathAD fom of the eult: NOTE: All paamete ae expeed in metic (SI) equialent alue fo dimenionle analyi ball popetie:.5 in D : R : m D D 0.05 R 0.09 coefficient of fiction between the cue ball and table cloth: μ : 0. appoximate alue fom eeal efeence (alo backed up by my own expeiment) aity : m 9.80 time equied fo cue ball to tat ollin (ceae lidin): in( ϕ) t( ω,, ϕ) co( ϕ) R ω : in ( ϕ) fom Equation 36 μ in( ϕ) elocity component when cue ball tat ollin in a taiht line: 5 xf (, ω, ϕ) : in( ϕ) co( ϕ) fom Equation 1 and 8 1 yf (, ω, ϕ) : 5 in( ϕ) R ω fom Equation and 9 the final deflected cue ball anle: 5 in( ϕ) co( ϕ) θ c (, ω, ϕ) : atan fom Equation 3, 8, and 9 5 in( ϕ) R ω
x poition of the cue ball duin the cued tajectoy: μ t co( ϕ) x c ( t,, ω, ϕ) : t in( ϕ) co( ϕ) fom Equation 34 co( ϕ) R ω in( ϕ) in( ϕ) x poition of the cue ball duin and afte the cued tajectoy: xt (,, ω, ϕ) : T t( ω,, ϕ) x c ( t,, ω, ϕ) if t T x c ( T,, ω, ϕ) xf (, ω, ϕ) ( t T) othewie y poition of the cue ball duin the cued tajectoy: μ t R ω in( ϕ) y c ( t,, ω, ϕ) t in( ϕ) in( ϕ) : fom Equation 35 co( ϕ) R ω in( ϕ) in( ϕ) y poition of the cue ball duin and afte the cued tajectoy: yt (,, ω, ϕ) : T t( ω,, ϕ) y c ( t,, ω, ϕ) if t T y c ( T,, ω, ϕ) yf (, ω, ϕ) ( t T) othewie Paamete ued in plot below: T : 5 numbe of econd to diplay t : 0, 0.01.. T 0.01 econd plottin incement ϕ : 30 de cut anle fo 1/-ball hit : 5 mph aeae peed in mph coneted to m/ m ω : oll peed R θ c (, ω, ϕ) 33.6 de deflected cue ball anle
Equation fo the tanent line: Equation fo the ball (fo cale) x tanent_line : t T x ball ( t) R co t π : T y tanent_line : x tanent_line ( t) tan( ϕ) y ball ( t) R in t π : T aiou follow hot with natual oll aiou peed (in mph, coneted to m/) fom low to ey fat: 1 : mph m : 4 mph m 3 : 6 mph m 4 : 8 mph m 1 3 4 ω 1 : ω R : ω R 3 : ω R 4 : R (,, ω (,, ω 3 (,, ω 4 yt, 1, ω 1, ϕ yt yt 3 yt 4 y tanent_line y ball 0.6 0.4 0. 0 0 0. 0.4 0.6 xt ( ) xt, 1, ω 1, ϕ,,, ω, ϕ, xt, 3, ω 3, ϕ, xt, 4, ω 4, ϕ, x tanent_line, x ball
aiou daw hot with eee natual oll 1 3 4 ω 1 : ω R : ω R 3 : ω R 4 : R 0.4 (,, ω (,, ω 3 (,, ω 4 yt, 1, ω 1, ϕ yt yt 3 yt 4 y tanent_line 0. 0 y ball y ball 0. 0.4 0 0. 0.4 0.6 0.8 xt ( ) xt, 1, ω 1, ϕ,,, ω, ϕ, xt, 3, ω 3, ϕ, xt, 4, ω 4, ϕ, x tanent_line, x ball
aiou lowe peed follow hot with natual oll 1 : 0.5 mph m : 1 mph m 3 : 1.5 mph m 4 : mph m 1 3 4 ω 1 : ω R : ω R 3 : ω R 4 : R (,, ω (,, ω 3 (,, ω 4 yt, 1, ω 1, ϕ yt yt 3 yt 4 0.4 0.3 y tanent_line 0. y ball 0.1 0 0 0.1 0. 0.3 0.4 xt ( ) xt, 1, ω 1, ϕ,,, ω, ϕ, xt, 3, ω 3, ϕ, xt, 4, ω 4, ϕ, x tanent_line, x ball