TP A.19 - Illustrated Principles of Pool and Billiards

technical proof
TP A.19
Masse shot aiming method, and curved cue ball paths
supporting:
“The Illustrated Principles of Pool and Billiards”
http://billiards.colostate.edu
by David G. Alciatore, PhD, PE ("Dr. Dave")
originally posted: 9/7/05 last revision: 2/19/11
The cue ball trajectory is based on the equations derived in TP A.4. See TP A.4 for
Equation number references and background information.
initial
cue ball
direction
(v )
yo
y
R
z
R A
a
A
F ~
top view
front view
c
x
x
final
cue ball
direction
(v )
f
F ~ F ~
a: offset right of center
b: offset below center
R: resting point on the table
A: impact line aim point on the table
z
b
R A
side view
y
technical proof

The cue ball is assumed to be struck in the y direction with the angled impulse as shown
above. From linear impulse and momentum, the initial cue ball speed components, after cue
stick impact, are:
vxo
0
~
F
v yo cos
m
v cos
where v e is the effective speed of a straight-on impact.
From angular impulse and momentum, the spin components of the cue ball about the x and y
axes, after cue stick impact, are related to the impulse according to:
~ 2
Fb I
2
xo mR
5
~
2
Fa sin 5
e
xo
2
Iyo mR yo
Therefore, the initial spin components, using Equation 2, can be written as:
5ve
xo b 2
5
2R
5
sin 2
2
a
ve R
( 1)
( 2)
( 3)
( 4)
( )
yo ( 6)
We can now find the final cue ball direction from Equation 23 in TP A.4:
tan
c
5v
5v
xo
yo
2R
2R
yo
xo
5ve
0 asin
R
asin
5ve
5v
b
R
e cos
cos
R
b As proved by Coriolis, the final cue ball direction can also be predicted from the geometry
shown in the illustration above. From the top view, the final cue ball direction can be
expressed as:
tan
a
c ____
RA
( 7)
( 8)

From the side view in the illustration above, it can be shown (see the figure below) that:
Equation 9 can be rearranged to give:
b RAsin
R cos
b
____
( 9)
RA
R
____ R cos
RA 10
sin
Using Equation 10 in Equation 8 results in the same expression as Equation 7. Therefore, the
aiming method proposed by Coriolis is shown to be valid. Note that the friction between the
ball and the table during cue stick impact is is not accounted for in this analysis, but Coriolis
also showed that this friction has no effect on the final cue ball direction.
With typcial conditions, the final CB angle usually comes up a little short of the angle
predicted by the Coriolis method. This could be due in part to squirt and multiple-body
collision effects associated with jamming the cue ball between the tip and slate.
Now we can define the cue ball trajectory based on the analyses above and in TP A.4.
NOTE: All parameters are expressed in metric (SI) equivalent values for dimensionless analysis
ball properties:
D
2.25in m
R
D
2
D 0.057
coefficient of friction between the cue ball and table cloth:
gravity
μ 0.2
g
g g
9.807
m
s 2
( )

From Equation 1, 2, 5, and 6 above,
vxo 0
ωxoveb 5v e
2R 2
b
vyoveϕ vecos( ϕ)
ωyovea ϕ
5v e
2R 2
asin( ϕ)
From Equation 14 in TP A.4,
vCoxvea b ϕ
vxo Rωyo vea ϕ
vCoyvea b ϕ
vyoveϕ Rωxo veb vCoxvea b ϕ
2
vCo vea b ϕ
From Equation 20 in TP A.4,
2v Co vea b ϕ
Δtvea b ϕ
7μg From Equations 21 and 22 in TP A.4,
2
vCoy vea b ϕ
vCox vea b ϕ
vxf vea b ϕ
vxo μg Δt
v
vCo vea b ϕ
ea b ϕ
vCoy vea b ϕ
vyf vea b ϕ
vyoveϕ μg Δt
v
vCo vea b ϕ
ea b ϕ
From Equations 24 and 25 in TP A.4,
vxot xc vea b ϕ t
ycvea b ϕ t
vyo veϕ
b ϕ
μgvCox vea b ϕ
t
2v Co ve a
2
t
b ϕ
μgvCoy vea b ϕ
t
2v Co ve a
2

From the analysis in TP A.4, the entire cue ball path is defined by:
xv ea b ϕ t
ΔT Δtvea b ϕ
xcvea b ϕ t
if t ΔT
xcvea b ϕ ΔT
vxf vea b ϕ(
t ΔT)
yv ea b ϕ t
ΔT Δtvea b ϕ
ycvea b ϕ t
if t ΔT
ycvea b ϕ ΔT
vyf vea b ϕ(
t ΔT)
From Equation 7 above, the final cue ball angle can be found with:
θc( ab ϕ)
angle( Rcos( ϕ)
basin(
ϕ)
)
Parameters used in plots below:
T 5
t 00.01 T
number of seconds to display
0.01 second plotting increment
otherwise
otherwise
Equation for the final direction: Equation for the ball (for scale)
xf () t
t
T 2 xball t ( ) Rcos t π
T 2
yf ( ab ϕ t)
xf () t
tanθc( ab ϕ)
yball( t)
R sin t π
T 2
various effective impact speeds (in mph, converted to m/s):
v1
6mph m
v2
9mph m
v3
s
s
12mph m
s

ϕt
yv
1
ab ϕt yv 2 a
0.6
yv
3
ab ϕt 0.4
y
f
( ab ϕt) y
ball
() t
0.2
b ϕt b ϕt yv
1
ab ϕt yv 2 a
yv 3 a
y
f
( ab ϕt) y
ball
() t
ϕ 75deg a 0.25R b 0.25R 0
0 0.2 0.4 0.6
0.2
0
0.2
0.4
xv 2
a
xv
1
abϕt bϕt xv 3
abϕt x f
() t x
ball
() t
ϕ 85deg a 0.25R b 0.25R 0 0.2 0.4 0.6
xv 2
a
xv
1
ab ϕt b ϕt xv 3
ab ϕt x f
() t x
ball
() t

Comparison of draw and follow shot swerve for a medium-fast-speed shot under
typical ball and table conditions:
v
6mph m
s
a 0.5Rcos( 45deg) bdraw a bfollow a
ϕdraw 4deg ϕfollow 3deg initial aiming line with a typical amount of squirt (2 degrees):
xaim() t
b follow
ϕ follow
t
yva b draw
ϕ draw
t
yva
y
aim
() t
t
y
T
aim() t
2
1.5
1
0.5
same amount of English for both:
1:30pm (follow) or 4:30pm (draw)
same amount of top or bottom spin for both
slightly higher cue elevation for the draw shot,
but typical values
xaim() t
tan( 2deg) 0
0 0.05 0.1 0.15 0.2
xva
xva b draw
ϕ draw
t
b follow
ϕ follow
t
x
aim
() t
The draw shot exhibits more "effective squirt" (the net effect of both squirt and swerve, AKA
squerve) over the shorter shot distances (0-4.5 ft); but, over longer shot distances (>4.5 ft), the
follow shot exhibits more "effective squirt." The crossover distance (4.5 ft) depends on shot
speed and ball/table conditions.
Here are the final cue-ball trajectory angles after sliding ceases and rolling begins:
2.193 deg
θc ab drawϕdraw
0.784 deg
θc ab followϕfollow

Final cue ball angle for various cue stick elevations:
θ
c
( ab ϕ)
deg
150
100
50
a 0.25R b 0.25R ϕ 0deg1 deg 90deg 0
0 20 40 60 80 100
ϕ
deg