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PROBLEM 12.107 (Continued)
PROBLEM 12.107 (Continued)
B
PROBLEM 12.108
Halley’s comet travels in an elongated elliptic orbit for which the minimum distance from the sun is
approximately
1
2,
E
r
where
6
150 10 km
E
r= ×
is the mean distance from the sun to the earth. Knowing that the
periodic time of Halley’s comet is about 76 years, determine the maximum distance from the sun reached by
the comet.
SOLUTION
max
PROBLEM 12.109
Based on observations made during the 1996 sighting of comet Hyakutake, it was concluded that the
trajectory of the comet is a highly elongated ellipse for which the eccentricity is approximately
0.999887.
ε
=
Knowing that for the 1996 sighting the minimum distance between the comet and the sun was
0.230 ,
E
R
where RE is the mean distance from the sun to the earth, determine the periodic time of the comet.
01 yr, (91.8 10 )(1.000)
PROBLEM 12.110
A space probe is to be placed in a circular orbit of radius 4000
km about the planet Mars. As the probe reaches A, the point of
its original trajectory closest to Mars, it is inserted into a first
elliptic transfer orbit by reducing its speed. This orbit brings it
to Point B with a much reduced velocity. There the probe is
inserted into a second transfer orbit by further reducing its
speed. Knowing that the mass of Mars is 0.1074 times the
mass of the earth, that rA = 9000 km and rB = 180,000 km, and
that the probe approaches A on a parabolic trajectory,
determine the time needed for the space probe to travel from A
to B on its first transfer orbit.
SOLUTION
6
PROBLEM 12.110 (Continued)
2ab
π
9
27.085 10
AB
×
AB
PROBLEM 12.111
A spacecraft and a satellite are at diametrically opposite positions
in the same circular orbit of altitude 500 km above the earth. As it
passes through point A, the spacecraft fires its engine for a short
interval of time to increase its speed and enter an elliptic orbit.
Knowing that the spacecraft returns to A at the same time the
satellite reaches A after completing one and a half orbits,
determine (a) the increase in speed required, (b) the periodic time
for the elliptic orbit.
SOLUTION
6
2
22
44
ππ
PROBLEM 12.111 (Continued)
66
0
9.0023 10 m, 6.87 10 m
A
a rr=×==×
66
2 11.1346 10 m, 8.7461 10 m
B A AB
r a r b rr= −= × = = ×
( )( )
66
92
3
2 9.0023 10 8.7461 10
258.159 10 m /s
8.5062 10
ab
h
π
π
τ
××
= = = ×
×
93
6
58.159 10 8.4656 10 m/s
6.87 10
A
A
h
vr
×
= = = ×
×
(a) Increase in speed at A.
33
08.4656 10 7.6119 10
AA
v vv∆= − = × − ×
854 m/s
A
v∆=
(b) Periodic time for elliptic orbit.
As calculated above 8.5062 s
τ
=
141.8 min
τ
=
PROBLEM 12.112
The Clementine spacecraft described an elliptic orbit of
minimum altitude
400
A
h=
km and a maximum altitude of
2940
B
h=
km above the surface of the moon. Knowing that
the radius of the moon is 1737 km and that the mass of the
moon is 0.01230 times the mass of the earth, determine the
periodic time of the spacecraft.
SOLUTION
6
9
3.78983 10
AB
h
×
4.95 h
τ
=
PROBLEM 12.113
Determine the time needed for the space probe of Problem 12.100 to travel from B to C.
SOLUTION
PROBLEM 12.113 (Continued)
2
1or
BA
A
BC
PROBLEM 12.114
A space probe is describing a circular orbit of radius nR with a velocity v0
about a planet of radius R and center O. As the probe passes through Point
A, its velocity is reduced from v0 to
b
v0, where
1,
b
<
to place the probe on a
crash trajectory. Express in terms of n and
b
the angle AOB, where B denotes
the point of impact of the probe on the planet.
SOLUTION
PROBLEM 12.115
A long-range ballistic trajectory between Points A and B on the earth’s
surface consists of a portion of an ellipse with the apogee at Point C.
Knowing that Point C is 1500 km above the surface of the earth and
the range
R
φ
of the trajectory is 6000 km, determine (a) the velocity
of the projectile at C, (b) the eccentricity
ε
of the trajectory.
SOLUTION
6
1.23548 cos153.016
+°
PROBLEM 12.115 (Continued)
PROBLEM 12.116
A space shuttle is describing a circular orbit at an altitude of 563 km above
the surface of the earth. As it passes through Point A, it fires its engine for
a short interval of time to reduce its speed by 152 m/s and begin its descent
toward the earth. Determine the angle AOB so that the altitude of the
shuttle at Point B is 121 km. (Hint: Point A is the apogee of the elliptic
descent trajectory.)
SOLUTION
2 6 2 12 3 2
PROBLEM 12.117
As a spacecraft approaches the planet Jupiter, it releases a probe which is to
enter the planet’s atmosphere at Point B at an altitude of 280 mi above the
surface of the planet. The trajectory of the probe is a hyperbola of eccentricity
Knowing that the radius and the mass of Jupiter are 44423 mi and
respectively, and that the velocity vB of the probe at B forms
an angle of 82.9° with the direction of OA, determine (a) the angle AOB,
(b) the speed of the probe at B.
SOLUTION
1.031.
ε
=
26
1.30 10 slug,×
B
v
33
B
PROBLEM 12.117 (Continued)
2
B
PROBLEM 12.118
A satellite describes an elliptic orbit about a planet. Denoting by
and the distances corresponding, respectively, to the perigee and
apogee of the orbit, show that the curvature of the orbit at each of
these two points can be expressed as
SOLUTION
0
r
1
r
01
1 11 1
2rr
ρ
= +
1cos A
GM C
2
2AB
rr
h
ρ
01
2rr
ρ
PROBLEM 12.119
(a) Express the eccentricity
ε
of the elliptic orbit described by a
satellite about a planet in terms of the distances r0 and r1
corresponding, respectively, to the perigee and apogee of the orbit.
(b) Use the result obtained in Part a and the data given in Problem
12.109, where to determine the approximate
maximum distance from the sun reached by comet Hyakutake.
SOLUTION
6
149.6 10 km,
E
R= ×
1(1 cos )
GM
11
E
PROBLEM 12.120
Derive Kepler’s third law of planetary motion from Eqs. (12.37) and (12.43).
22
a
τ
