Game Development Reference
In-Depth Information

()
()
x
t
+
kt
x
=
g
(13.90)
The method of undetermined coefficients provides the following particular solu-
tion to Equation (13.90).
g
()
(13.91)
x
t
=
t
k
Adding the general solution to the homogeneous differential equation, we have
g
()
kt
x
t
=+
AB
e
+
t
,
(13.92)
k
where the vectors A and B are arbitrary constants that can be determined by es-
tablishing initial conditions. Specifying the initial position x and initial velocity
v , we have
()
()
x
0
0
=
=
x
0
x
v
.
(13.93)
0
()
x at time
()
Setting these equal to the values given by the functions
x
t
and
t
=
t
0
gives us the system
ABx
g Bv ,
+=
0
k
+=
(13.94)
0
k
from which we can derive the following expressions for A and B .
g
v
0
Ax
=− +
0
2
k
k
gv
0
B
=−
(13.95)
k
2
k
()
The position function
x
t
for an object moving through a resistive medium is
given by
k
v
g
g
(
)
()
0
x
t
=++
x
t
1
e
kt
.
(13.96)
0
2
k
k
()
()
The velocity function
v
t
is given by the derivative of
x
t
: