Game Development Reference
In-Depth Information
v
n1
w
r
p
ˆ
F
n
Figure 6-15.
A frictional impulse causes a rotation.
It is assumed that the object is not rotating when it collides with the ramp, so the initial
angular velocity,
w
0
, is equal to zero. The angular impulse, frictional impulse, and final angular
velocity,
w
1
, are related by Equation (6.30).
ˆ
=
ˆ
F
r
=
Iw
1
(6.34)
The quantity,
r
, is the distance from the center of mass of the object to the point of contact.
The negative sign appears on the right-hand side of Equation (6.34) because a positive frictional
impulse causes a negative angular velocity. The same frictional force appears in both
Equations (6.33) and (6.34), so the two equations can be combined into a single equation that
relates the change in velocity normal to the line of action to the change in angular velocity.
I
r
ω
(
)
mv
−=−
v
1
(6.35)
n
1
n
0
There are two unknowns in Equation (6.35),
v
n
1
and
w
1
. In order to solve for the two
unknowns, one more equation that relates them is needed. This expression can be obtained if
we consider the nature of frictional impulses. A frictional impulse only exists as long as there is
friction—in other words, as long as the ball is sliding over the ramp. When the translational and
angular velocities reach the point where the ball starts rolling without sliding, the frictional
impulses no longer act upon the golf ball. The condition for rolling without sliding is when the
velocity normal to the line of action,
v
n
1
, is equal to the product of the angular velocity,
w
1
, and
the distance,
r
.
v
=
rω
(6.36)
n
1
1
Using Equations (6.35) and (6.36), expressions can be derived for the velocity normal to
the line of action and the angular velocity.
v
v
=
n
0
n
1
I
mr
(6.37)
1
+
2
