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.

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