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

:

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