Game Development Reference
In-Depth Information
The location of an object is a vector quantity, and Equation (4.6) can be divided into directional
components. In the Cartesian frame of reference, Equation (4.6) would be split into x-, y-, and
z-direction components.
dx
dy
dz
v
=
v
=
v
=
(4.7)
x
y
z
dt
dt
dt
Since acceleration is the derivative of velocity with respect to time and velocity is the derivative
of location with respect to time, it follows that acceleration is the second derivative (the
derivative of the derivative) of location with respect to time.
⎛
⎟
2
ddx
dx
⎜
a
=
=
⎟
(4.8)
⎜
⎜
⎝⎠
⎟
x
2
dt
dt
dt
It is helpful to look at the relationship between acceleration, velocity, and location visu-
ally. Figure 4-1 shows the acceleration, velocity, and location (altitude in this case) when an
object is dropped (or skis off a cliff) and falls vertically towards the ground. The acceleration
due to gravity is constant. Because the acceleration is constant, the velocity increases linearly
with time. Since the speed of the object is continually increasing, location of the object decreases
along a parabolic-shaped curve.
Figure 4-1.
The acceleration, velocity, and location curves of a falling object
Recall from Chapter 1 that a derivative can be thought of as the slope of a curve. The slope
of the location curve at a given time is equal to the velocity. The slope of the velocity curve at a
given time is equal to the acceleration.
Solving the Translational Equations of Motion
Newton's second law and the derivatives presented in the previous section provide the basic
equations needed to solve for the acceleration, velocity, and location of an object as a function
