Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 524

The first observation is that the projectile reaches its maximum height,
denoted x max , when the acceleration has consumed all of the velocity and
v(t) = 0. It's easy to solve for the time when this will occur by using
Equation (11.13), v(t) = v 0 + at:

Time to reach apex

v(t) = 0,
v 0 + at = 0,
t = −v 0 /a.

Right now we are in one dimension and considering only the height. But
if we are in more than one dimension, only the velocity parallel to the
acceleration must vanish. There could be horizontal velocity, for example.
We discuss projectile motion in more than one dimension in just a moment.
The second observation is that the time it takes for the object to travel
from its maximum altitude to its initial altitude, denoted t e in Figure 11.12,
is the same as the time taken to reach the maximum. In other words, the
projectile reaches its apex at t e /2.
The third and final observation is that the velocity at t = t e , which we
have denoted v e , has the same magnitude as the initial velocity v 0 , but the
opposite sign.
Before we look at projectile motion in more than one dimension, let's
summarize the formulas we have derived in this section. The first two are
the only ones worth memorizing; the others can be derived from them.

Summary of Kinematics Equations Dealing with Constant Acceleration

v(t) = v 0 + at,
∆x = v 0 t + (1/2)at 2 ,
x(t) = x 0 + ∆x = x 0 + v 0 t + (1/2)at 2 ,
v 0 = ∆x/t − (1/2)at,

−v 0 ± v 0 + 2a∆x

a

t = , (11.17)

a = 2 ∆x − v 0 t
t 2

.

Extending the ideas from the previous section into 2D or 3D is mostly
just a matter of switching to vector notation; x, v, and a become p , v , and