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