Game Development Reference
In-Depth Information
mv
(
−+
v
)
m v
(
− =
v
)
0
(6.12a)
11
x
1
x
2 2
x
2
x
vv
=
(6.12b)
1
y
1
y
vv
=
(6.12c)
2
y
2
y
If you look at Equation (6.12), you will see that momentum is conserved in both the x- and
y-directions, but only the velocities in the x-direction change as a result of the collision. If you
recall, the line of action for this analysis was taken to be parallel to the x-axis. This observation
highlights a general rule about linear collision analysis: a linear collision will only change the
velocities in the direction of the line of action of the collision . The velocity components normal
to the line of action of a collision will be unchanged by the collision. Strictly speaking, these
conclusions are only valid if there is no friction between the colliding objects, but that is currently
the assumption for our analysis.
Looking more closely at the velocity expressions in Equation (6.12), there is still a problem.
There are four unknowns, , but only three equations. In order to solve for the
post-collision velocities, an additional equation is introduced that relates the relative pre- and
post-collision velocities of the two spheres along the line of action of the collision.
vvv v
,
,
,
11 2 2
x
y
x
y
(
)
(
)
ev
−=−−
v
v
v
(6.13)
1
x
2
x
1
x
2
x
The coefficient, e , is known as the coefficient of restitution and has a value between 0 and 1.
The coefficient of restitution relates back to the earlier discussion on elastic and inelastic collisions.
If e = 1, the pre- and post-collision relative velocities are equal, meaning that the collision is
elastic. On the other hand, if e = 0, the post-collision relative velocity is zero (meaning that the
objects are stuck together) and the collision is completely inelastic. As you might expect, in
most situations the coefficient of restitution will have a value between 0 and 1.
Combining Equations (6.12a) and (6.13), expressions can be obtained for the post-collision
velocities along the line of action.
(
)
1
+
em
mem
=
2
v
1
2
v
+
v
(6.14a)
1
x
1
x
2
x
mm
+
mm
+
1
2
1
2
(
)
1
+
em
mem
1
v
=
v
+
2
1
v
(6.14b)
2
x
1
x
2
x
mm
+
mm
+
1
2
1
2
We can see from Equation (6.14) that the post-collision velocities along the line of action
of the collision are a function of the pre-collision velocities along the line of action, the masses
of the two objects, and the coefficient of restitution. The velocities in the y-direction, perpen-
dicular to the line of action of the collision, are unaffected by the collision.
The development in this section has assumed a two-body collision in which the line of
action of the collision is parallel to the x-axis. Of course, this situation will not always be the case.
In general, the vector that defines the line of action for a collision can have any orientation.
You'll learn how to handle general, two-, and three-dimensional collisions a little later in the
chapter.
 
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