Game Development Reference

In-Depth Information

or whether or not it is traveling at a constant 60 mi/hr. It could very well be that the car

was accelerating (or decelerating) over that 30 m distance.

To more precisely analyze the motion of the car in this example, you need to understand

the concept of
instantaneous
velocity. Instantaneous velocity is the specific velocity at

a given instant in time, not over a large time interval as in the car example. This means

that you need to look at very small Δ
t
's. In math terms, you must consider the limit as

Δ
t
approaches 0—that is, as Δ
t
gets infinitesimally small. This is written as follows:

v = lim
Δt→0
(Δs/Δt)

In differential terms, velocity is the derivative of displacement (change in position) with

respect to time:

v = ds/dt

You can rearrange this relationship and integrate over the intervals from
s
1
to
s
2
and
t
1

to t
2
, as shown here:

v dt = ds

∫
(s1 to s2)
ds = ∫
(t1 to t2)
v dt

s2 - s1 = Δs = ∫
(t1 to t2)
v dt

This relation shows that displacement is the integral of velocity over time. This gives

you a way of working back and forth between displacement and velocity.

Kinematics makes an important distinction between displacement and distance trav‐

eled. In one dimension, displacement is the same as distance traveled; however, with

vectors in space, displacement is actually the vector from the initial position to the final

position without regard to the path traveled, while displacement is the difference be‐

tween the starting position coordinates and the ending position coordinates. Thus, you

need to be careful when calculating average velocity given displacement if the path from

the starting position to the final position is not a straight line. When Δ
t
is very small (as

it approaches 0), displacement and distance traveled are the same.

Another important kinematic property is acceleration, which should also be familiar to

you. Referring to your driving experience, you know that acceleration is the rate at which

you can increase your speed. Your friend who boasts that his brand new XYZ 20II can

go from 0 to 60 in 4.2 seconds is referring to acceleration. Specifically, he is referring to

average acceleration.

Formally, average acceleration is the rate of change in velocity, or Δ
v
over Δ
t
:

a = Δv/Δt