Game Development Reference
In-Depth Information
The Relationship Between Force, Acceleration, Velocity,
and Location
Newton's second law, which was introduced in Chapter 3, relates the net external force on an
object,
F
, to the mass and acceleration of the object.
Fma
=
(4.1)
The arrows above the force and acceleration terms in Equation (4.1) indicate that they are
vector quantities having both a magnitude and a direction. The force and acceleration vectors
can be divided into components that act in the individual coordinate directions. For example, in
the Cartesian reference frame, force and the resulting accelerations can be divided into x-, y-,
and z-components.
F
=
a
F
=
a
F
=
a
(4.2)
y
y
x
x
z
z
One of the motivations for splitting the force and acceleration vectors into directional
components is that the directional components can be analyzed independently of each other.
A force in the x-direction will have no effect on acceleration in the y-direction. If the net force
acts in only one direction, it can simplify the modeling of a problem in that the other two direc-
tions can oftentimes be ignored.
Once the accelerations are known, the velocity of the object can be determined. Acceleration
is the time rate of change of velocity. Recall from Chapter 1 that a time rate of change of some-
thing can be expressed as a derivative. Acceleration can be represented as the derivative of
velocity,
v
, with respect to time.
dv
a
=
(4.3)
dt
As indicated in Equation (4.3), velocity is a vector quantity and can be split into directional
components. In the Cartesian frame of reference, the relationship between acceleration and
velocity derivative could be divided into x-, y-, and z-components.
dv
dv
dv
y
a
=
x
a
=
a
=
z
(4.4)
x
y
z
dt
dt
dt
While velocity can be expressed in terms of its three directional components, there are
times when the velocity magnitude, or speed, is required. Velocity magnitude,
v
, can be found
by obtaining the square root of the sum of the square of the directional components.
v
=++
v
2
v
2
v
2
(4.5)
x
y
z
If you have taken geometry, you may recognize Equation (4.5) as the Pythagorean theorem.
Once the velocity of an object has been determined, the location of an object can be
computed from the velocity components. Velocity is the rate of change of location,
s
, with
respect to time.
ds
v
=
(4.6)
dt
