Game Development Reference
In-Depth Information
g
g
()
()
kt
v
t
==+−
x
t
v
e
.
(13.97)
0
k
k
Over time, the velocity of an object whose motion is being resisted ap-
proaches a constant called the terminal velocity . The terminal velocity
v is given
()
by the limit of the velocity function
v
t
as t tends to infinity:
g
()
v
=
lim
v
t
=
.
(13.98)
T
k
t
→∞
Although it is not apparent from Equation (13.96), the position function for
an object moving through a resistive medium does converge to the familiar Equa-
tion (13.74) as the constant k approaches zero. This can be seen by evaluating the
limit
gvg
k
(
)
()
0
x
t
=
lim
x
+
t
+
1
e
kt
.
(13.99)
0
2
k
k
k
0
Replacing the exponential function with its power series (see Appendix D, Equa-
tion (D.11)), we have
22
33
44
gvg
k
k t
k t
k t

()
0
x
t
=
lim
x
+
t
+
kt
+
+ −
0
2
k
k
2!
3!
4!
k
0

2
3
2
4
g
t
t
t
kt

(
)
=
lim
x
+
t
+
k
v
g
+
+ −
0
0
k
k
2!
3!
4!
k
0

3
2
4
kt
k t

2
2
(
)
=
lim
xv
+
t
1
kt
v
+
1
g
t
+
k
vg
+ −
0
0
0
0
2
2
3!
4!

k
0
2
(13.100)
=+ +
xv
t
1
g
t
.
0
0
2
13.5 Friction
Friction is the well-known force that arises when two surfaces are in contact. We
discuss two types of friction in this section: kinetic friction and static friction.
Kinetic friction occurs between two surfaces that are in motion relative to each
other and has the effect of resisting that motion. Static friction refers to the force
that holds a stationary object in place when it is in contact with another surface.