TP B.8 Draw shot physics - Billiards and Pool Principles ... · From TP 2.1, the safe miscue limit...

technical proof

technical proof

TP B.8Draw shot physics

supporting:“The Illustrated Principles of Pool and Billiards”

http://billiards.colostate.edu

by David G. Alciatore, PhD, PE ("Dr. Dave")

originally posted: 2/21/2009 last revision: 7/10/2011

R

vs

b

v

ms

mb

v’

’

ddrag

’’

dskid

vOB

v’’’

’’’=v’’’/R

droll

ddraw

stopped

v’’

In the analysis below, I will begin by assuming the balls and tip are perfect (eb=1.0, μb=0,

ηtip=100%), but I will account for these effects later. With perfect balls, v''=0, ω''=-ω', and

vOB=v'.

Relevant physical parameters (from "physics" FAQ):

μs 0.2 typical ball-cloth coefficient of sliding friction

μr 0.01 typical ball-cloth coefficient of rolling resistance

R 1.125 in ball radius:

mr6

19 typical ball-mass-to-cue-mass ratio (mb/ms):

eb 1 ball-to-ball coefficient of restitution (assumed perfectly elastic)

μb 0.06 ball-to-ball coefficient of friction (varies with ball surface speed)

From TP A.30, the CB speed (v) and spin (ω) resulting from cue speed v s and tip offset b,

assuming a perfect tip, are:

v vs b 2 vs

1 mr5

2

b

R

2

ω vs b 5

2 v vs b

b

R2

Note - ω<0 implies backspin; ω>0 implies topspin or forward roll

From TP 2.1, the safe miscue limit is generally assumed to be:

bmaxR

20.563 in b 0 0.05 bmax bmax

Based on TP B.6, typical draw shot CB speeds (which are higher than other shots) are probablyin the following range:

vslow 8 mph vmedium 10 mph vfast 14 mph vpower 19 mph

From the equation above for ball speed, these correspond to center-ball-hit cue speeds of:

vs v( )1 mr

2v

vs_medium vs vmedium 6.579 mphvs_slow vs vslow 5.263 mph

vs_fast vs vfast 9.211 mph vs_power vs vpower 12.5 mph

Here is how the cue ball speed varies with tip offset for a given cue speed, assuming a perfecttip:

0 20 40 60 80 100100

110

120

130

140

150

160

v vs_medium b vs_medium %

b

bmax %

v vs_medium 0 vs_medium

152 %

v vs_medium bmax vs_medium

103.051 %

So with a center-ball hit, assuming a perfect tip, the CB speed is about 50% morethan (3/2-times) the cue speed; and at a maximum offset shot, the CB andoriginal cue speeds are nearly equal.

Here is how the cue ball spin varies with tip offset for a given cue speed:

0 20 40 60 80 1000

500

1 103

1.5 103

ω vs_medium b

rpm

b

bmax %

ω vs_medium bmax ω vs_medium 0.5bmax

151.695 %


104.313 %

So, assuming a perfect tip, more tip offset always gives you more spin, but theincrease in spin isn't as great with increasing offset. The spin at maximumoffset is only about 50% faster than the spin at half the maximum offset, and onlyabout 5% than the spin at 90% maximum offset.

From TP 4.1, the CB speed and spin change during drag over distance x on theway to the OB according to:

vdrag v x μs v2

2 μs g x

ωdrag v ω x μs ω5

2 Rv v

22 μs g x

From TP 4.1, the distance required for stun to develop (for ω < 0) is:

dstun v ω μs 2 R ω( )

5 μs gv

ω( ) R

5

From TP B.5, the distance required for a skidding (sliding) CB to develop natural roll, andthe speed when roll first develops are:

dskid v ω μs signv

Rω

2

49 μs g 6 v

2 5v R ω R ω( )

2

vskid v ω( )5

7v

2

7R ω

From TP B.6, the distance required for a rolling ball to come to a stop is:

droll v μr sign v( )v

2

2 μr g

and the speed slows during rolling over distance x according to:

vroll v x μr v2

2 μr g x

For a draw shot, where the CB still has backspin at OB impact,

ddraw dskid v'' ω'' μs droll v''' μr =

Assuming perfect balls (v''=0, ω''=-ω'),

ddraw dskid 0 ω' μs droll v''' μr =

ω'' ω'= ωdrag v ω ddrag μs =

v v vs b =

ω ω vs b =

v''' vskid 0 ω' μs =

Combining everything gives:

ddraw dskid 0 ω' μs droll v''' μr =

ddraw2 R ω'( )

2

49 μs g

v'''2

2 μr g=

ddraw

2 R ωdrag v ω ddrag 2

49 μs g

vskid 0 ω'( )2

2 μr g=

ddraw2

49 μs gR ωdrag v ω ddrag 2

1

2 μr g

2

7R ωdrag v ω ddrag

2

=

ddraw2R

2

49g

1

μs

1

μr

ωdrag v ω ddrag 2=

ddraw2R

2

49g

1

μs

1

μr

ω5

2 Rv v

22 μs g ddrag

2

=

To account for energy loss in the tip-ball collision, we can use the following equations for ballspeed and spin instead (from TP A.30), which account for tip inefficiency:

η 0.87 typical cue tip efficiency

v vs b vs

1 η1 η

mr1

5

2

b

R

2

1 mr5

2

b

R

2

ω vs b 5

2 v vs b

b

R2

Here is how the cue ball speed varies with tip offset for a given cue speed, accounting for tipinefficiency:

0 20 40 60 80 10060

80

100

120

140

v vs_medium b vs_medium %

b

bmax %

v vs_medium 0 vs_medium

127.452 %

v vs_medium bmax vs_medium

74.628 %

So with a center-ball hit, and a typical tip inefficiency, the CB speed is about30% more than the cue speed; and at a maximum tip offset shot, the CB speed isabout 75% (3/4) of the original cue speed.

Here is how the cue ball spin varies with tip offset for a given cue speed:

0 20 40 60 80 1000

200

400

600

800

1 103

ω vs_medium b

rpm

b

bmax %


134.983 %


100.728 %

So, with a typical tip inefficiency, more tip offset gives you more spin, but theincrease in spin isn't as great with increasing offset. The spin at maximumoffset is only about 35% faster than the spin at half the maximum offset. Also,the spin at 90% offset is nearly equal to the spin at maximum (100%) offset.

To account for ball inelasticity, from TP A.5 (Equation 7),

v''1 eb

2v'=

and to account for ball friction, from TP A.27,

μ v( ) 9.951 103

0.108 e

1.088v

m

s

vrel v ωx ωz ϕ v sin ϕ( ) R ωz 2 R ωx cos ϕ( ) 2

tanθ v ωx ωz ϕ min

μ vrel v ωx ωz ϕ v cos ϕ( )

vrel v ωx ωz ϕ 1

7

v sin ϕ( ) R ωz

v cos ϕ( )

Δω v ωx ωz ϕ 5

2

v

R tanθ v ωx ωz ϕ

where Δω is the change in spin experienced by both the CB and OB.

With this analysis, topspin in analogous to sidespin in TP A.27, and the throw is in the verticaldirection. Therefore,

ϕ 0 ωz ω'= ωx 0

no cut angle (topspin) no sidespin (English)

With topspin, ωz > 0 and Δω < 0. With bottomspin, ωz < 0 and Δω > 0.

For a follow shot, the CB spin, after impact, is:

ω'' ω' Δω=

For a draw shot, the CB spin, after impact, is:

ω'' ω' Δω( )=

Now let's see how draw distance varies with cue speed, tip offset, shot distance, andtable conditions. The following program calculates the draw distance for any case,and accounts for imperfect balls and an imperfect tip.

ddraw vs b ddrag μs μr v v vs b

ω ω vs b

dstun dstun v ω μs

"CB hits OB with topspin"

dpre_roll dskid v ω μs

"CB rolls before OB contact"

vpre_roll vskid v ω( )

d_roll_stop droll vpre_roll μr

"CB doesn't reach OB"

v' 0

d_roll_stop ddrag dpre_roll if

"CB rolls into OB"

v' vroll vpre_roll ddrag dpre_roll μr

otherwise

ω'v'

R

dpre_roll ddrag if

"CB slides into OB with partial roll"

ω' ωdrag v ω ddrag μs

v' vdrag v ddrag μs

otherwise

v''1 eb

2v'

ω'' ω' Δω v' 0 ω' 0( )( )

v''' vskid v'' ω''( )

ddraw dskid v'' ω'' μs droll v''' μr

dstun ddrag if

"CB hits OB with backspin"

v' vdrag v ddrag μs

ω' ωdrag v ω ddrag μs

ω'' ω' Δω v' 0 ω' 0( )( )

v''1 eb

2v'

v''' vskid v'' ω''( )

ddraw dskid v'' ω'' μs droll v''' μr

otherwise

Typical shot distances:

ddrag_short 1 ft ddrag_medium 4 ft ddrag_long 8 ft

Valid range of tip offset (for no miscue): b 0 inbmax

50 bmax

NOTE - In all of the plots below, a negative ddraw implies follow and a zero implies a stop shot

or the CB not making it to the OB.

Draw distance vs. tip offset for various cue speeds:

ddrag ddrag_short

0 20 40 60 80 100

0

10

20

30

0

ddraw vs_slow b ddrag μs μr ft

ddraw vs_medium b ddrag μs μr ft

ddraw vs_fast b ddrag μs μr ft

ddraw vs_power b ddrag μs μr ft

b

bmax %

A

ddrag ddrag_medium

0 20 40 60 80 100

0

10

20

0





b

bmax %

B

ddrag ddrag_long

0 20 40 60 80 10010

5

0

5

10

0





b

bmax %

C

Draw distance vs. tip offset for various shot (drag) distances:

vs vs_slow

0 20 40 60 80 1006

4

2

0

2

4

0

ddraw vs b ddrag_short μs μr ft

ddraw vs b ddrag_medium μs μr ft

ddraw vs b ddrag_long μs μr ft

b

bmax %

D

vs vs_medium

0 20 40 60 80 10010

5

0

5

10

0




b

bmax %

E

vs vs_fast

0 20 40 60 80 10010

0

10

20

0




b

bmax %

F

vs vs_power

0 20 40 60 80 10010

0

10

20

30

40

0




b

bmax %

G

Sensitivity analysis for cue speed and tip offset for various draw distances:

ddrag ddrag_mediumb 0 0.5% bmax 50 % bmax

0 10 20 30 40 501

0

1

2

0

1ddraw vs_fast b ddrag μs μr ft

ddraw 1.25vs_fast b ddrag μs μr ft

b

bmax %

H

ddrag ddrag_short b 0 0.5% bmax 15 % bmax

0 5 10 150.2

0.1

0

0.1

0.2

0




b

bmax %

I

ddrag ddrag_medium b 10% bmax 10.1% bmax 30 % bmax

10 15 20 25 300

0.05

0.1

0.15

0.2

0

.1



b

bmax %

J

ddrag ddrag_medium

vs_low vs_fast vs_high 1.5 vs_fast vs vs_low vs_low

vs_high vs_low

50 vs_high

bsmall 25% bmax bmedium 50% bmax blarge 100 % bmax

9 10 11 12 13 140

5

10

15

0

ddraw vs bsmall ddrag μs μr ft

ddraw vs bmedium ddrag μs μr ft

ddraw vs blarge ddrag μs μr ft

vs

mph

K

v1 0.75 vs_fast v2 vs_fast v3 1.25 vs_fast

b1 24.6% bmax b2 14.5% bmax b3 10% bmax

stun-through (roll through)stun-follow shot: ddrag 1.5 ft 1.51 ft 2.5 ft

1.4 1.6 1.8 2 2.2 2.4 2.60

0.05

0.1

0.15

0

ddraw v1 b1 ddrag μs μr ft



ddrag

ft

L

stun-back (roll back)stun-draw shot:

b1 12.5% bmax b2 5.9% bmax b3 3.2% bmax

1.4 1.6 1.8 2 2.2 2.4 2.60.1

0.08

0.06

0.04

0.02

0 0




ddrag

ft

M

stop shot:

b1 17% bmax b2 9.3% bmax b3 5.87% bmax

1.4 1.6 1.8 2 2.2 2.4 2.6

0.02

0

0.02

0




ddrag

ft

N

ddrag 3.5 ft 3.51 ft 4.5 ft

v1 0.75 vs_power v2 vs_power v3 1.25 vs_power

b1 18.5% bmax b2 9.9% bmax b3 6.2% bmax

3.4 3.6 3.8 4 4.2 4.4 4.60.02

0.01

0

0.01

0.02

0




ddrag

ft

O

Let's look at how draw distance varies with shot distance, speed, and tip offset for a fairlypowerful draw shot, where the OB is 6 feet away and you want to draw the CB 6 feetback to its original location:

ddrag 6 ft ddrag 0.75 ddrag 0.76 ddrag 1.25 ddrag

v1 11.65 mph v2 11.25 mph v3 11.1 mph v4 12.42 mph

b1 bmax b2 .9 bmax b3 .75 bmax b4 .5 bmax

5.6 5.8 6 6.2 6.45

5.5

6

6.5

7

6





6

ddrag

ft

P

ddrag 6 ft b 0 0.5% bmax 100 % bmax

0 20 40 60 80 1005

0

5

10

15

ddraw v1 b ddrag μs μr ft


b

bmax %

Q

Let's look at how draw distance varies with shot distance, speed, and tip offset for anexample draw shot, where the OB is 9 feet away and you want to draw the CB theequivalent of 18 feet back:




8 8.5 9 9.5 1015

16

17

18

19

20

18





ddrag

ft

R


0 20 40 60 80 100

0

10

20

30



b

bmax %

S

Let's look at how draw distance varies with shot distance, speed, and tip offset for a moremodest draw shot, where the OB is 3 feet away and you want to draw the CB 3 feet backto its original location:




2 2.5 3 3.5 40

2

4

6

3

6





ddrag

ft

T


0 20 40 60 80 1005

0

5

10



b

bmax %

U

Now let's look at the effects of cloth conditions (a sticky cloth vs. a slick cloth):

μs_slick 0.1 μs_sticky 0.3

vs1 10 mph vs2 15 mph

b1 36.6% bmax b2 80% bmax b3 22.9% bmax b4 30.1% bmax

ddrag 2 ft 2.1 ft 5ft

2 3 4 50

2

4

6

0

ddraw vs1 b1 ddrag μs_slick μr ft

ddraw vs1 b2 ddrag μs_sticky μr ft

ddraw vs2 b3 ddrag μs_slick μr ft

ddraw vs2 b4 ddrag μs_sticky μr ft

ddrag

ft

V

Optimal tip offset for maximum draw for different speeds and drag distances:

vs vs_power ddrag ddrag_long b .75 bmax (initial guess)

b_opt b( ) ddraw vs b ddrag μs μr

Given

b 0

b bmax

b_opt Maximize b_opt b( ) 0.436 in

b_opt

bmax77.48 % ddraw vs b_opt ddrag μs μr 6.84 ft

Here are some example values:

vs_slow 5.263 mph ddrag_short 1 ft b_opt 82.3% d_draw 2.6ft

vs_medium 6.579 mph ddrag_short 1 ft b_opt 87.7% d_draw 6.2ft

vs_fast 9.211 mph ddrag_short 1 ft b_opt 92.7% d_draw 16.2ft

vs_power 12.5 mph ddrag_short 1 ft b_opt 95.2% d_draw 33.9ft

vs_medium 6.579 mph ddrag_medium 4 ft b_opt 61.8% d_draw 0.0ft

vs_fast 9.211 mph ddrag_medium 4 ft b_opt 77.9% d_draw 5.3ft

vs_power 12.5 mph ddrag_medium 4 ft b_opt 87.7% d_draw 24.6ft

vs_fast 9.211 mph ddrag_long 8 ft b_opt 61.1% d_draw 0.0ft

vs_power 12.5 mph ddrag_long 8 ft b_opt 76.3% d_draw 8.1ft

Useful conclusions from all of the plots above, which are consistent with generallyaccepted "best practices" and good-player intuition:

Generally, to get more draw, you must hit the cue ball harder and lower (see1.graphs A and G). No big surprise here!

More tip offset does not produce significantly more draw as you approach the2.miscue limit; so, generally, it is advisable to not hit too close to the miscuelimit (see graphs A and G).

With larger drag distances, and for a given maximum cue speed, max draw3.occurs at less than maximum tip offset (at about 70%-80% tip offset). In otherwords, you don't get more power draw by hitting closer to the miscue limit (seeGraphs G, Q, S, and U, and see the data on the previous page).

In general, with a draw shot with a medium desired draw distance, a slower cue4.speed with more tip offset will result in better draw distance control than afaster cue speed with less offset (see the slopes of the curves in Graphs A and Gat a given draw distance).

Stop shots are much less sensitive to tip offset position than draw shots are. In5.other words, CB position is much easier to control with a stop shot , ascompared to a draw shot (see Graphs G, H, I, N, and O).

For a short stop shot, slower speed offers slightly better control (see the overall6.slopes of the curves in Graph I). For longer stop shots, faster speed appears tooffer slightly better control (see Graphs N and O).

For stun-through (small controlled follow) and stun-back (small controlled draw),7.a firmer hit closer to center offers better CB control (see Graphs L and M).

It is much easier to control draw distance on a new, slick cloth than it is on a8."sticky" cloth, especially with lower-speed shots (see Graph T). The statementassumes the player is equally well "adjusted" to each cloth condition. Any playerwill need to adjust when playing under different cloth conditions.

It is easier to draw the ball on slick cloth, and faster cloth allows for9.greater draw distances .

TP B.8 Draw shot physics - Billiards and Pool Principles ... · From TP 2.1, the safe miscue limit...

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Transcript of TP B.8 Draw shot physics - Billiards and Pool Principles ... · From TP 2.1, the safe miscue limit...