Graphics Programs Reference
In-Depth Information
EXAMPLE 7.8
The aerodynamic drag force acting onacertain object in free fall can be approxi-
matedby
av
2
e
−
by
F
D
=
where
v
=
velocity of the object in m/s
y
=
elevation of the object in meters
a
=
7
.
45 kg/m
10
−
5
m
−
1
b
=
10
.
53
×
The exponentialterm accounts for the changeofair densitywith elevation. The dif-
ferentialequationdescribing the fall is
my
=−
mg
+
F
D
80665 m/s
2
and
m
where
g
114kgis the mass of the object. If the object is
releasedat an elevation of 9 km, determine its elevation and speedaftera10s fall with
the adaptive Runge-Kuttamethod.
=
9
.
=
Solution
The differentialequation and the initialconditions are
a
m
y
2
exp(
y
=−
g
+
−
by
)
7
45
114
.
y
2
exp(
10
−
5
y
)
=−
9
.
80665
+
−
10
.
53
×
y
(0)
=
9000 m
y
(0)
=
0
Letting
y
1
=
y
and
y
2
=
y
, weobtain the equivalent first-order equations and the
initialconditions as
y
1
y
2
y
2
+
65
10
−
3
y
2
exp(
y
=
=
10
−
5
y
1
)
−
9
.
80665
.
351
×
−
10
.
53
×
9000 m
0
y
(0)
=
The functiondescribing the differentialequations is
Search WWH ::
Custom Search