TP A.20 The effect of spin, speed, and cut angle on draw shots

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1 technical proof TP A.2 The effect of pin, peed, and cut angle on draw hot technical proof upporting: The Illutrated Principle of Pool and Billiard by David G. Alciatore, PhD, PE ("Dr. Dave") originally poted: 11/8/5 lat reviion: 11/8/5 Refer to TP A.4 for the background derivation and illutration. ball geoetry: D := 2.25 in R := D 2 D =.57 R =.29 coefficient of friction between the cue ball and table cloth: µ := gravity g := g 2 g = 9.87 tie required for the cue ball to tart rolling (ceae liding): 2v t vω,, := in 7 µ g 2 R ω in( ) + + v in( ) co 2 velocity coponent when the cue ball tart rolling in a traight line: 5 v xf v, ω, := 1 := ( 2 2R ω) v yf v, ω, 7 v in 7 5v in co( )

2 the final deflected cue ball angle: := atan2( 5 v in( ) 2 2R ω, 5v in( ) co( ) ) θ c v, ω, x poition of the cue ball during the curved trajectory: := vt x c tv,, ω, in co( ) 2 co µ g t 2 co( ) 2 R ω in( ) + + v in( ) 2 x poition of the cue ball during and after the curved trajectory: := T t( vω,, ) if t T xtv,, ω, x c tv,, ω, + v xf ( v, ω, ) ( t T) otherwie x c Tv,, ω, y poition of the cue ball during the curved trajectory: := vt y c tv,, ω, 2 in µ g t 2 in 2 co R ω + v in( ) 2 R ω in( ) + + v in( ) 2 y poition of the cue ball during and after the curved trajectory: := T t( vω,, ) if t T ytv,, ω, y c tv,, ω, + v yf ( v, ω, ) ( t T) otherwie y c Tv,, ω, Equation for the tangent line: Equation for the ball (for cale): x tangent_line y tangent_line t, := t T 2 x ball ( t) R co t π := T 2 := x tangent_line t tan ( t) R in t π := T 2

3 Plot data: := := 1 in( ) f 3 v := 5 ph v ω roll := R f_range :=, ( f) := ain( 1 f) cut angle for 1/2-ball hit average peed in ph converted to / average natural roll pin rate ball-hit fraction range cut angle and ball-hit fraction relationhip (fro TP 3.3) SRF_range :=, T := 5 t :=,.1.. T pin rate factor range (fro TP A.12) nuber of econd to diplay.1 econd plotting increent variou peed (in ph, converted to /), fro low to very fat: v 1 := 2 ph v 2 := 4 ph v 3 := 6 ph v 4 := 8 ph natural roll follow hot (ee TP 3.3): θ c v, ω roll, ( f_range ) f_range

4 natural (revere natural roll) draw hot: 2 15 θ c v, ω roll, ( f_range ) 1 5 For a 1/2-ball hit: half := 3 θ c v, ω roll, half f( half ) =.5 = f_range Cut angle (and ball-hit fraction) required for the deflected cue ball direction to be perpendicular to the original (aiing line) direction: 9 := 3 initial gue (1/2-ball hit) Given (, ω roll, ) 9 = 9 θ c v 9 := Find 9 9 = f( 9 ) =.368 Aount of draw (pin rate factor) required for perpendicular deflection at a 1/2-ball hit: 9 := SRF := 1 3 f( 9 ) =.5 initial gue (natural roll) Given θ c v, SRF ω roll, 9 SRF_TYP := = 9 Find( SRF) SRF_TYP =.625 Thi will be referred to a a "typical aount of pin for a draw hot." It i 62.5% of the natural roll rate.

5 variou peed 1/2 ball-hit draw hot with typical aount of pin: v 1 v 2 v 3 ω 1 := SRF_TYP ω R 2 := SRF_TYP ω R 3 := SRF_TYP R.4 (,, ω 2, ) (,, ω 3, ) ytv, 1, ω 1, ytv 2 ytv 3 y tangent_line ( t, ) xtv (, 2, ω 2, ) xtv, 1, ω 1,,, xtv, 3, ω 3,, x tangent_line, x ball

6 variou ball-hit fraction draw hot with typical aount of pin: θ c_biect := 18 ω := SRF_TYP ω roll Double biect draw hot aiing yte decribed in Jewett' October '95 BD article: 4 Triect aiing yte propoed here: θ c_alt := θ c ( v, ω, ( f_range) ) θ c_biect ( ( f_range) ) θ c_alt ( ( f_range) ) f_range 1/4-ball hit: 1 := (.25) 1 = = θ c v, ω, 1 θ c_biect 1 = θ c_alt ( 1 ) = /2-ball hit: 2 := (.5) 2 = 3 3/4-ball hit: = 9 θ c v, ω, 2 3 := (.75) θ c_biect ( 2 ) = 6 3 = θ c_alt ( 2 ) = 9 θ c v, ω, 3 7/8-ball hit: = = θ c v, ω, 4 4 := 7 8 θ c_biect ( 3 ) = = θ c_biect ( 4 ) = θ c_alt ( 3 ) = θ c_alt ( 4 ) = The triec yte work fairly well over a wide range of cut angle (ball-hit fraction). The double biect yte i better for all cut angle (large ball-hit fraction). Both yte are poor for large cut angle (all ball-hit fraction), epecially the double biect ethod.

7 .4 (,, ω, ) 2 (,, ω, ) 3 ytv,, ω, 1 ytv ytv xtv xtv,, ω, 1,,, ω, 2, xtv,, ω, 3, x ball

8 half-ball hit draw hot at variou pin rate: = 3 1/2-ball hit 12 θ c v, SRF_range ω roll, SRF_range 75% of typical draw hot pin (25% le than typical): SRF 1 :=.75 SRF_TYP SRF 1 =.469 ω 1 := SRF 1 ω roll = θ c v, ω 1, typical draw hot pin: SRF 2 := SRF_TYP SRF 2 =.625 ω 2 := SRF 2 ω roll = 9 θ c v, ω 2, 125% of typical draw hot pin (25% ore than typical): SRF 3 := 1.25 SRF_TYP SRF 3 =.781 ω 3 := SRF 3 ω roll = θ c v, ω 3,

9 .4 (,, ω 2, ) (,, ω 3, ) ytv,, ω 1, ytv ytv y tangent_line ( t, ) xtv (,, ω 2, ) xtv,, ω 1,,, xtv,, ω 3,, x tangent_line, x ball

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