Game Development Reference

In-Depth Information

F
IGURE
2.8

Same average velocity, different instantaneous velocity.

accurate picture of the velocity of the object at one instant in time. If we could make

this gap infinitely small, then we would have the exact answer.

In mathematical notation this is written using the “limit” notation

p

t

v

=

lim

t

→

0

which simply means that the velocity would be accurately given by the distance trav-

eled divided by the time gap (
p/t
), if we could make the gap infinitely small

(lim
t
→
0
). Rather than use the cumbersome limit notation, this is more commonly

written with a lowercase d in place of the
:

p

t
=

d
p

d
t

v

=

lim

t

→

0

Because it is so common in mechanics to be talking about the change with respect to

time, this is often simplified even further:

d
p

d
t

=
˙

v

=

p

The dot over the
p
simply means that it is the velocity at which
p
is changing—its

differential with respect to time.

Acceleration

If
p
is the position of an object and
v
is its velocity (where
v

=
˙

p
), we can define its

acceleration too.

Acceleration is the rate at which velocity is changing. If an object is accelerat-

ing quickly, it is rapidly increasing in velocity. In normal English we use the word

slowing
to mean the opposite of acceleration, or
braking
if we are talking about a

car. In physics acceleration it can mean any change in velocity, either increasing or

decreasing velocity. A positive value for acceleration represents speeding up, a zero