Game Development Reference

In-Depth Information

acceleration means no change in velocity at all, and a negative acceleration represents

slowing down.

Because acceleration represents the rate at which velocity is changing, following

the same process we arrive at

v

t

d
v

d
t

a

=

lim

t

=

→

0

where
v
in this formula is a velocity itself, also defined in terms of its own limit, as

wesawearlierinthissection.Wecouldwritethisas
a

=˙

v
if we wanted to, but this

causes problems.

As long as I use
v
for velocity, it is fairly clear, but in general if I write

d
m

d
t

it would not be obvious whether
m
is a velocity (and therefore the whole expression

is an acceleration) or if it is a position (making the expression a velocity). To avoid

this confusion, it is typical to write accelerations in terms of the position only.

This is called the “second differential.” Velocity is the first differential of position,

and if we differentiate again, we get acceleration; so acceleration is the second differ-

ential. Mathematicians often write it in this way:

d
2
p

d
t
2

d
v

d
t
=

d

d
t

d
p

d
t
=

a

=

which can be confusing if you're not used to differential notation. Don't worry about

how we end up with the strange set of squared things—it isn't important for us; it

simply indicates a second differential. Fortunately we can completely ignore this for-

mat altogether and use the dotted form again—

d
2
p

d
t
2

=
¨

a

=

p

which is the format we'll use in the remainder of the topic.

We could carry on and find the rate at which the acceleration is changing (this is

called the “jerk” or sometimes the “jolt,” and it is particularly important in the de-

sign of roller coasters, among other things). We could go further and find the rate of

change of jerk, and so on. It turns out, however, that these quantities are not needed

to get a believable physics engine running. As we shall see in the next chapter, Newton

discovered that applying a force to an object changes its acceleration: to make believ-

able physics involving forces, all we need to keep track of are the position, velocity,

and acceleration.

So, in summary:
p
, the velocity of
p
, is measured at one instant of time, not at an

average velocity; and

¨

p
is the acceleration of
p
, measured in exactly the same way, and

it can be negative to indicate slowing down.