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PROBLEM 12.113 (Continued)
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
v0, where 1,
to place the probe on a
crash trajectory. Express in terms of n and
the angle AOB, where B denotes
the point of impact of the probe on the planet.
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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
of the trajectory is 6000 km, determine (a) the velocity
of the projectile at C, (b) the eccentricity
of the trajectory.
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.)
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.
1.031.
26
1.30 10 slug,
B
v
PROBLEM 12.117 (Continued)
2(1 cos )
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
0
r
1
r
01
1111
2rr
PROBLEM 12.119
(a) Express the eccentricity
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.
6
149.6 10 km,
E
R
PROBLEM 12.120
Derive Kepler’s third law of planetary motion from Eqs. (12.37) and (12.43).
