Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 516

The accelerations experienced by an object can vary as a function of
time, and indeed we can continue this process of differentiation, resulting
in yet another function of time, which some people call the “jerk” function.
We stick with the position function and its first two derivatives in this
book. Furthermore, it's very instructive to consider situations in which
the acceleration is constant (or at least has constant magnitude). This is
precisely what we're going to do in the next few sections.
Section 11.6 considers objects under constant acceleration, such as ob-
jects in free fall and projectiles. This will provide an excellent backdrop to
introduce the integral, the complement to the derivative, in Section 11.7.
Then Section 11.8 examines objects traveling in a circular path, which ex-
perience an acceleration that has a constant magnitude but a direction that
changes continually and always points towards the center of the circle.

11.6 Motion under Constant Acceleration

Let's look now at the trajectory an object takes when it accelerates at a
constant rate over time. This is a simple case, but a common one, and
an important one to fully understand. In fact, the equations of motion we
present in this section are some of the most important mechanics equations
to know by heart, especially for video game programming.
Before we begin, let's consider an even simpler type of motion—motion
with constant velocity. Motion with constant velocity is a special case of
motion with constant acceleration—the case where the acceleration is con-
stantly zero. The motion of a particle with constant velocity is an intuitive
linear equation, essentially the same as Equation (9.1), the equation of a
ray. In one dimension, the position of a particle as a function of time is

x(t) = x 0 + vt, (11.14)

where x 0 is the position of the particle at time t = 0, and v is the constant
velocity.
Now let's consider objects moving with constant acceleration. We've
already mentioned at least one important example: when they are in free
fall, accelerating due to gravity. (We'll ignore wind resistance and all other
forces.) Motion in free fall is often called projectile motion. We start out
in one dimension here to keep things simple. Our goal is a formula x(t) for
the position of a particle at a given time.
Take our example of illegal ball-bearing-bombing off of Willis Tower.
Let's set a reference frame where x increases in the downward direction,
and x 0 = 0. In other words, x(t) measures the distance the object has
fallen from its drop height at time t. We also assume for now that initial