Game Development Reference
In-Depth Information
Linear Momentum and Impulse
In Chapter 2, you learned about force, mass, acceleration, and velocity and used these concepts to
model the motion of an object. In modeling a collision between two objects, another physical
quantity needs to be introduced—
momentum
. Momentum is another way to characterize the
state of an object in motion.
The linear momentum,
p
, of an object is simply the mass of the object,
m
, multiplied by its
velocity,
v
.
pmv
=
(6.1)
As introduced in Chapter 2, the arrow symbols in Equation (6.1) indicate vector quantities.
Linear momentum has units of
kg-m/s
in the SI system of units or
slug-ft/s
under the English
system. Since velocity is a vector quantity, momentum is as well. Under the Cartesian coordinate
system, the overall linear momentum of an object can be separated into x-, y-, and z-components.
pmv
=
(6.2a)
x
x
p
=
v
(6.2b)
y
y
pmv
=
(6.2c)
z
z
In Chapter 3, Newton's second law of motion was introduced, which relates the net external
force on an object to a resulting acceleration of the object. As shown in Equation (6.3), Newton's
second law can also be written in terms of a velocity derivative.
==
dv
Fmam
dt
(6.3)
If the mass of an object is constant, it can be moved into the derivative term, and Newton's
second law can be written in terms of momentum.
==
dmv
dp
F
(6.4)
dt
dt
Equation (6.4) states that the change of the momentum with respect to time is equal to the
net external force applied to the object. If the net force on the object is zero, then the derivative
of momentum with respect to time is zero—which means that momentum is constant. To
determine the change in momentum caused by an applied force, the left- and right-hand sides
of Equation (6.4) can be integrated with respect to time.
−=
∫
pp
t
(6.5)
1
0
In Equation (6.5),
p
0
is the initial value of linear momentum and
p
1
is the momentum at the
end of the time interval being considered. Equation (6.5) indicates that a change in momentum
of an object is equal to the integral of the net external force on the object as a function of time.
The integration of the force with respect to time is known as the
linear impulse
of force.
ˆ
F
∫
=
t
(6.6)
