Graphics Programs Reference
In-Depth Information
EXAMPLE 7.6
R
e
v
0
r
H
A spacecraft islaunchedat an altitude
H
=
772 kmabovesea level with the speed
v
0
=
6700 m/s in the direction shown. The differentialequations describing the mo-
tion of the spacecraft are
2
r
˙
θ
GM
e
r
2
2
r
˙
¨
r
=
θ
−
θ
=−
r
where
r
and
are the polar coordinates of the spacecraft. The constants involvedin
the motionare
θ
10
−
11
m
3
kg
−
1
s
−
2
G
=
6
.
672
×
=
universal gravitationalconstant
10
24
M
e
=
5
.
9742
×
kg
=
mass of the earth
R
e
=
6378
.
14km
=
radius of the earth atsea level
(1) Derive the first-orderdifferentialequations and the initialconditions of the form
y
=
,
=
b
. (2) Use the fourth-order Runge-Kuttamethod to integrate the
equationsfrom the timeoflaunch until the spacecraft hits the earth.Determine
F
(
t
y
),
y
(0)
θ
at
the impact site.
Solution of Part (1)
Wehave
GM
e
=
6
10
−
11
5
10
24
=
10
14
m
3
s
−
2
.
672
×
.
9742
×
3
.
9860
×
Letting
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
y
1
y
2
y
3
y
4
r
r
˙
y
=
θ
the equivalent first-order equations become
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
y
1
y
2
y
3
y
4
y
1
y
0
y
3
10
14
y
0
−
3
.
9860
×
/
y
=
y
3
−
2
y
1
y
3
/
y
0
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