Game Development Reference
In-Depth Information
Combining Equations (6.5) and (6.6) results in an equation relating a linear impulse of
force to the resulting change in linear momentum.
=−=
ˆ
Fpp mvv
(
)
−
(6.7)
1
0
1
0
Let's apply Equation (6.7) to the collision of a moving object with something solid such as
a wall. When the object strikes the wall, it will change the direction of its flight and therefore its
velocity components will change as well. According to Equation (6.7), this change in velocity is
the result of a linear impulse of force acting on the object due to the collision. The time of the
collision, or
dt
, is generally very small, so according to Equation (6.6), in order for the impulse
to be large enough to significantly change the momentum of the object, the force acting on the
object must be very large. The force due to collision is known as an
impulsive force
. The magnitude
of the impulsive force is usually so much larger than any other forces (gravity, drag, etc.) acting
on the object during the collision, that all other forces can be ignored during the collision.
An important feature of impulsive force and linear impulse of force is that they act normal
to the point of impact. As we shall see in a little while, the change in velocity due to a collision
occurs normal to the point of impact as well. Newton's third law applies to linear impulses. If
one object exerts a linear impulse on another, the second object will exert an equal and oppo-
site impulse on the first object.
Conservation of Linear Momentum
According to the momentum version of Newton's second law shown in Equation (6.4), if there
is no net force applied to an object, its momentum will remain constant. In other words, if
there is no net force, momentum is conserved. It is also clear from Equation (6.5) that if there
is no net force applied to an object over a certain time interval, the momentum of the object
remains constant over that time interval.
What is true for a single object is also true for two or more objects being analyzed as a
group, also called a
system of objects
. For instance, if the system under consideration consists
of two objects and if no net forces are applied to the objects, then the linear momentum of the
system of objects will remain constant.
mv
+
m v
=
C
(6.8)
11
2 2
The
C
parameter in Equation (6.8) is some constant value. Equation (6.8) states that the
overall linear momentum of the two-body system is constant if no external forces are acting on
the objects, but linear momentum conservation also applies to the directional components of
momentum. If there are no net forces in the x-, y-, and z-directions, then linear momentum is
conserved in the x-, y-, and z-directions.
mv
+
m v
=
C
(6.9a)
11
x
2 2
x
x
mv
+
m v
=
C
(6.9b)
11
y
2 2
y
y
mv
+
m v
=
C
(6.9c)
11
z
2 2
z
z
