Game Development Reference
In-Depth Information
Combining Equations (7.6) and (7.7) results in an expression for the post-collision velocity
of the golf ball normal to the line of action of the collision. As before, the result is independent
of the coefficient of friction between the ball and club face.
mv
mv
sin
a
2
2
′
v
=
ccn
= −
ccx
(7.8)
bn
7(
mm
+
)
7 (
mm
+
)
c
b
c
b
One thing to keep in mind about the preceding development is that it assumes the golf ball
eventually rolls without sliding along the club face. This assumption is good for lower-loft
angle clubs. At higher loft angles, 45 degrees and up, the ball may never reach this point and
will be in a combined state of sliding and rolling when it leaves the club face.
The last step in the golf ball collision analysis is to convert the post-collision velocities of
the golf ball parallel and normal to the line of action back to the standard Cartesian x- and z-
components,
v
bx
and
v
bz
. This operation can be performed using the inverse rotation matrix
shown in Equation (6.19).
′
′
′
v
=
v
cos
α
−
v
sin
α
(7.9)
bx
bp
bn
vv
′
=
′
sin
α
+
v
′
cos
α
(7.10)
bz
bp
bn
If you want to get fancy, Equations (7.3), (7.8), (7.9), and (7.10) can be combined into
expressions that relate the post-collision x- and z-velocity components of the golf ball to the
initial velocity, ball mass, club head mass, coefficient of restitution, and loft angle.
⎛
⎟
m
2
⎜
′
=
(
)
2
2
v
v
c
1c s
+
e
α
+
sin
α
⎟
⎜
(7.11)
⎝
⎟
bx
cx
⎠
(
mm
+
)
7
c
b
mm
α
⎛
⎟
m
5
⎜
′
=
v
v
c
sin
cos
+
⎟
e
⎜
(7.12)
⎝
⎟
bz
cx
⎠
(
+
)
7
c
b
A Sample Collision Analysis
The equations developed in the previous section may seem rather complicated, but the process
for computing the post-collision velocities of a golf ball is really pretty straightforward. Let's go
through an example to see how it all fits together so far. A driver with a club head mass of 0.2
kg
and a loft angle of 10 degrees strikes a two-piece ball with an impact velocity of 50
m/s
. Assume
the coefficient of restitution between the ball and the club is 0.78. We want to compute the
post-collision velocities and spin rate of the ball.
The first step in the process is to compute the velocity components normal and parallel to
the line of action for the collision. The parallel and normal velocities of the club head,
v
cp
and
v
cn
, can be obtained from Equations (7.1) and (7.2).
10
p
⎛
⎞
m
s
v
=
50 cos
=
49.24
(7.13)
⎜
⎟
cp
180
⎝
⎠
10
p
⎛
⎞
m
s
v
=−
50 sin
=−
8.68
(7.14)
⎜
⎟
cn
180
⎝
⎠
