Some Basic Concepts - Physics for Game Programmers - page 16

Figure 2-6. The altitude of the snowboarder as a function of time

You might think of modeling the rate of change of altitude of the snowboarder over time
by simply subtracting the final altitude and time from the initial altitude and time.

z − z

Δ z

t

final initial

=

(2.18)

Δ t − t

final initial

form shown in Equation (2.18) is the equivalent of drawing a straight line from
the endpoints of the curve shown in Figure 2-6. Equation (2.18) provides the average rate of
change of altitude with time. However, as shown in Figure 2-7, it fails to accurately calculate
the rate of fall at a given time, because, unlike the curve shown in Figure 2-6, this result is linear.
At first the snowboarder's rate of fall is small. When he hits the ground, his rate of fall is much
larger. The

Using the Δ

expression shown in Equation (2.18) only correctly predicts the altitude of the
snowboarder at the beginning and end of his fall.
A derivative, on the other hand, provides an instantaneous rate of change. In the case of
the snowboarder, it's the rate of change in altitude at any time during the fall of the snowboarder.
Instead of using the

Δ

symbol, derivatives are represented using the letter d . A derivative that
represents the change in altitude over time would be written as the following:

Δ

dz
dt

(2.19)

The variable on top of the derivative, z in this case, is known as the dependent variable ,
because its value depends on changes to the variable on the bottom of the derivative known as
the independent variable . The derivative shown in Equation (2.19) would be referred to as “the
derivative of z with respect to t .”