TP A.28 Throw plots for all types of shots · Throw plots for all types of shots supporting: ......

technical proof

technical proof

TP A.28Throw plots for all types of shots

supporting:“The Illustrated Principles of Pool and Billiards”

http://billiards.colostate.eduby David G. Alciatore, PhD, PE ("Dr. Dave")

originally posted: 4/3/2007 last revision: 9/21/2007

Note: These results are based on the analysis in TP A.14.

R1.125 in⋅

m:= ball radius converted to meters

Model of friction based on Marlow data (Table 10 on p. 245 in "The Physics of Pocket Billiards," 1995)

vd1 .1 sin 45 deg⋅( )⋅:= μd1 0.11:=

vd2 1 sin 45 deg⋅( )⋅:= μd2 0.06:=

vd3 10 sin 45 deg⋅( )⋅:= μd3 0.01:=

We can solve for the coefficients from the set of data above using:

initial guesses:

a μd3:= b μd1 μd3−:= c1−

vd2ln

μd2 a−

b

⋅:=

a 0.01= b 0.1= c 0.98=

Given

μd1 a b ec− vd1⋅

⋅+=


⋅+=


⋅+=

a

b

c

Find a b, c, ( ):=

a

b

c

9.951 10 3−×

0.108

1.088

=

μ v( ) a b e c− v⋅⋅+:=

this is the "model" of friction that will be used

MathCAD formulations of Equations 15 through 17, using the above formulation for friction:

vrel v ωx, ωz, ϕ, ( ) v sin ϕ( )⋅ R ωz⋅−( )2 R ωx⋅ cos ϕ( )⋅( )2+:=

θthrow v ωx, ωz, ϕ, ( ) atan

min

μ vrel v ωx, ωz, ϕ, ( )( ) v⋅ cos ϕ( )⋅

vrel v ωx, ωz, ϕ, ( )17

v sin ϕ( )⋅ R ωz⋅−( )⋅

v cos ϕ( )⋅

:=

cut angle range:ϕ 0 deg⋅ 1 deg⋅, 75 deg⋅..:=

various speeds (slow, medium, fast):

v1 0.5ms

⋅:= v2 1.5ms

⋅:= v3 4.5ms

⋅:=

v1 1.118 mph= v2 3.355 mph= v3 10.066 mph=

v1v1m

s

:= v2v2m

s

:= v3v3m

s

:=

spin rate from percent English (from TP A.25):

ω v PE, ( ) SRF ωroll⋅=54

vR

⋅ PE⋅=

ω v PE, ( )54

vR

⋅ PE⋅:=

throw angle equation variable info

θthrow v ωx, ωz, ϕ, ( )

ωx, ωr: vertical plane spin rate (follow or draw) ωz, ωe: English spin rate

PEr: % topspin (roll) PEe: % English (side spin)

topspin: ωr, PEr > 0 outside English (OE): ωe, PEe > 0

bottom-spin: ωr, PEr < 0 inside English (IE): ωe, PEe < 0

These plots show how the amount of throw varies with the amount of English for various cutangles, speeds, and amounts of vertical plane spin (draw or follow):

PEr 0%:= ϕ 0 deg⋅:= PEe 100− %⋅ 99− %, 100 %⋅..:=

100− 0 100

5−

0

5

θthrow v1 ω v1 PEr, ( ), ω v1 PEe, ( ), ϕ, ( )deg



PEe

%PEr 0%:= ϕ 30 deg⋅:=

100− 0 100

5−

0

5




PEe

%

PEr 0%:= ϕ 60 deg⋅:=

100− 0 100

5−

0

5




PEe

%PEr 50%:= ϕ 0 deg⋅:=

100− 0 100

5−

0

5




PEe

%

PEr 50%:= ϕ 30 deg⋅:=

100− 0 100

5−

0

5




PEe

%

PEr 50%:= ϕ 60 deg⋅:=

100− 0 100

5−

0

5




PEe

%

PEr 100%:= ϕ 0 deg⋅:=

100− 0 100

5−

0

5




PEe

%

PEr 100%:= ϕ 30 deg⋅:=

100− 0 100

5−

0

5




PEe

%

PEr 100%:= ϕ 60 deg⋅:=

100− 0 100

5−

0

5




PEe

%

The remaining plots show how throw varies with cut angle for various speed shots with various amounts ofEnglish and vertical plane spin (follow or draw):

PEr 0%:= PEe 0%:= ϕ 0 deg⋅ 1 deg⋅, 75 deg⋅..:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 25%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 50%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 75%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 100%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 25− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 50− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 75− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 0%:= PEe 100− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 0%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 25%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 50%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 75%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 100%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 25− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 50− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 75− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 50%:= PEe 100− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 0%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 25%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 50%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 75%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 100%:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 25− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 50− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 75− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

PEr 100%:= PEe 100− %:=

0 15 30 45 60 75

5−

0

5




ϕ

deg

TP A.28 Throw plots for all types of shots · Throw plots for all types of shots supporting: ......

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Transcript of TP A.28 Throw plots for all types of shots · Throw plots for all types of shots supporting: ......