Game Development Reference
In-Depth Information
Figure 2-7.
Dz/Dt does not accurately predict the rate of fall at a given time.
The derivative
dz/dt
can also be thought of as the slope of the curve shown in Figure 2-6,
and its value will change over time. When the snowboarder starts his fall, the slope of the altitude
vs. time curve is shallow, and the value of the derivative is a small negative number (it's negative
because his altitude is decreasing with time). At the end of his fall, the rate of descent is large
and so are the slope of the curve and the value of the derivative.
The derivative shown in Equation (2.19) represents the rate of change of one scalar quan-
tity to another scalar quantity. This type of derivative is known as a
first-order derivative
. It's
possible to take a derivative of another derivative. A derivative of a derivative is called a
second-
order derivative
and is indicated by using the number 2 as superscripts in the derivative
expression. For example, the z-component of velocity is defined as the derivative of altitude
with respect to time.
dz
v
=
(2.20)
z
dt
Acceleration is defined as the derivative of velocity with respect to time, but acceleration
can also be written as the second derivative of altitude with respect to time.
⎛
⎟
2
dv
ddz
dz
⎜
a
==
z
=
⎟
(2.21)
⎜
⎜
⎝⎠
⎟
z
2
dt
dt
dt
dt
To see graphically the relationship between first and second derivatives, the altitude, velocity,
and acceleration of the unfortunate snowboarder are shown as a function of time in Figure 2-8.
The acceleration and velocity values are really negative, but they are shown as positive values
in Figure 2-8 to make the figure more compact.
