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Problems 299

TABLE 9.3 THE FIRST ARTIFICIAL EARTH SATELLITES

MEAN DISTANCE
FROM CENTER OF EARTH PERIGEE APOGEE
SATELLITE MASS (SEMIMAJOR AXIS) DISTANCE DISTANCE PERIOD

3
3
3
Sputnik I 83 kg 6.97 10 km 6.60 10 km 7.33 10 km 96.2 min
Sputnik II 3000 7.33 6.61 8.05 104
Explorer I 14 7.83 6.74 8.91 115
Vanguard I 1.5 8.68 7.02 10.3 134
Explorer III 14 7.91 6.65 9.17 116
Sputnik III 1320 7.42 6.59 8.25 106

41. Calculate the orbital periods of Sputnik I and Explorer I from 9.5 Energy in Orbital Motion
their apogee and perigee distances in Table 9.3.
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42. The aphelion distance for Saturn is 1510 10 km; its perihelion 45. The Voskhod I satellite, which carried Yuri Gagarin into space
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distance is 1350 10 km. By Kepler’s First Law, the Sun is at in 1961, had a mass of 4.7 10 kg. The radius of the orbit
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was (approximately) 6.6 10 km. What were the orbital
one focus of this ellipse. How far from the Sun is the other focus?
speed and the orbital energy of this satellite?
How does this compare with the orbital radius of Mercury?
46. What is the kinetic energy and what is the gravitational
43. The comet Hale–Bopp was spectacularly visible in the spring
potential energy for the orbital motion of the Earth around
of 1997 (see Fig. 9.37) and may be the most viewed comet in
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history. Its perihelion distance was 137 10 km, and its orbital the Sun? What is the total energy?
period is 2380 yr. What is its aphelion distance? How does this 47. Compare the escape velocity given by Eq. (9.27) with the
compare with the mean distance of Pluto from the Sun? velocity required for a circular orbit of radius R , according to
S
Eq. (9.10). By what factor is the escape velocity larger than the
velocity for the circular orbit?
48. In July of 1994, fragments of the comet Shoemaker–Levy
struck Jupiter.
(a) What is the impact speed (equal to the escape speed) for a
fragment falling on the surface of Jupiter?
(b) What is the kinetic energy at impact for a fragment of
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1.0 10 kg? Express this energy as an equivalent number
of short tons of TNT (the explosion of 1 short ton, or
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2000 lb, of TNT releases 4.2 10 J).
49. A 1.0-kg mass is in the same orbit around the Earth as the
Moon (but far from the Moon). What is the kinetic energy for
this orbit? The gravitational potential energy? The total energy?
50. The boosters on a satellite in geosynchronous orbit acciden-
tally fire for a prolonged period. At the instant this “burn”
FIGURE 9.37 Comet Hale–Bopp photographed in 1997. ends, the velocity is parallel to the original tangential direc-
tion, but the satellite has been slowed to one-half of its origi-
nal speed. The satellite is thus at apogee of its new orbit.
What is the perigee distance for such an orbit? What happens
44. The orbit of the Earth deviates slightly from circular: at to the satellite?
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aphelion, the Earth–Sun distance is 1.52 10 km, and 51. A black hole is so dense that even light cannot escape its grav-
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at perihelion it is 1.47 10 km. By what factor is the itational pull. Assume that all of the mass of the Earth is com-
speed of the Earth at perihelion greater than the speed pressed in a sphere of radius R. How small must R be so the
at aphelion? escape speed is the speed of light?