Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 519

Figure 11.10
Graphical representation of Table 11.3
procedure will work for any arbitrary velocity function. 28 This limit argu-
ment is a formalized tool in calculus known as the Riemann integral, which
we will consider in Section 11.7. That will also be the appropriate time to
consider the general case of any v(t). However, since there is so much we
can learn from this specific example, let's keep it simple as long as possible.
Remember the question we're trying to answer: how far does an object
travel after being dropped at an initial zero velocity and then accelerated
due to gravity for 2.4 seconds at a constant rate of 32 ft/s 2 ? How does
this new realization about the equivalence of distance traveled and the area
under the graph of v(t) help us? In this special case, v(t) is a simple linear
function, and the area under the curve from t = 0 to t = 2.4 is a triangle.
That's an easy shape for us to compute an area. The base of this triangle
28 Well, almost. There are certain limitations we must place on v(t). For example, if it
blows up and goes to infinity, it's likely, though not certain, that the displacement will
be infinite or undefined. In this topic, we are focused on physical phenomena and so we
sidestep these issues by assuming our functions will be well-behaved.