Page 100 - Fisika Terapan for Engineers and Scientists

300 CHAPTER 9 Gravitation

52. The spectacular comet Hale–Bopp (Fig. 9.37), most visible in melt the material of the satellite? To vaporize it? The heats
1997, entered the Solar System in an elliptical orbit with of fusion and of vaporization of aluminum are given in
period 4206 yr. However, after a close encounter with Jupiter Table 20.4.
on its inbound path, it continues on a new elliptical orbit with *57. According to one theory, glassy meteorites (tektites) found on
a period of 2380 yr. By what fraction did the encounter with the surface of the Earth originate in volcanic eruptions on the
Jupiter change the energy of Hale–Bopp’s orbit? Moon. With what minimum speed must a volcano on the
53. The typical speed of nitrogen molecules at a temperature of Moon eject a stone if it is to reach the Earth? With what
117 C, the temperature of the Moon’s surface at “noon,” is speed will this stone strike the surface of the Earth? In this
600 m/s; some molecules move slower, others faster. What problem ignore the orbital motion of the Moon around the
fraction of the escape velocity from the Moon is this? Can Earth; use the data for the Earth–Moon system listed in the
you guess why the Moon has not retained an atmosphere? tables printed inside the book cover. (Hint: When the rock
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*54. The Andromeda galaxy is at a distance of 2.1 10 m from reaches the intermediate point where the gravitational pulls of
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our Galaxy. The mass of the Andromeda is 6.0 10 kg, and the Moon and the Earth cancel out, it must have zero velocity.)
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the mass of our Galaxy is 4.0 10 kg. *58. A spacecraft is launched with some initial velocity toward the
(a) Gravity accelerates the galaxies toward each other. As reck- Moon from 300 km above the surface of the Earth.
oned in an inertial reference frame, what is the acceleration (a) What is the minimum initial speed required if the space-
of the Andromeda galaxy? What is the acceleration of our craft is to coast all the way to the Moon without using its
Galaxy? Treat both galaxies as point particles. rocket motors? For this problem pretend that the Moon
(b) The speed of the Andromeda galaxy relative to our Galaxy does not move relative to the Earth. The masses and
is 266 km/s. What is the speed of the Andromeda and radii of the Earth and the Moon and their distance are
what is the speed of our Galaxy relative to the center of listed in the tables printed inside the book cover. (Hint:
mass of the two galaxies? The center of mass is at a point When the spacecraft reaches the point in space where the
between the galaxies such that the distances of the galax- gravitational pulls of the Earth and the Moon cancel, it
ies from this point are in the inverse ratios of their masses. must have zero velocity.)
(c) What is the kinetic energy of each galaxy relative to the (b) With what speed will the spacecraft strike the Moon?
center of mass? What is the total energy (kinetic and 59. The Pons–Brooks comet had a speed of 47.30 km/s when it
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potential) of the system of the two galaxies? Will the two reached its perihelion point, 1.160 10 km from the Sun. Is
galaxies eventually escape from each other? the orbit of this comet elliptical, parabolic, or hyperbolic?
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*55. Neglect the gravity of the Moon, neglect atmospheric friction, *60. At a radial distance of 2.00 10 m from the center of the
and neglect the rotational velocity of the Earth in the follow- Earth, three artificial satellites (I, II, III) are ejected from a
ing problem. A long time ago, Jules Verne, in his book From rocket. The three satellites I, II, III are given initial speeds of
Earth to the Moon (1865), suggested sending an expedition to 5.47 km/s, 4.47 km/s, and 3.47 km/s, respectively; the initial
the Moon by means of a projectile fired from a gigantic gun. velocities are all in the tangential direction.
(a) With what muzzle speed must a projectile be fired verti- (a) Which of the satellites I, II, III will have a circular orbit?
cally from a gun on the surface of the Earth if it is to Which will have elliptical orbits? Explain your answer.
(barely) reach the distance of the Moon? (b) Draw the circular orbit. Also, superimposed on the same
(b) Suppose that the projectile has a mass of 2000 kg. What diagram, draw the elliptical orbits of the other satellites;
energy must the gun deliver to the projectile? The explo- label the orbits with the names of the satellites. (Note:
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sion of 1 short ton (2000 lb) of TNT releases 4.2 10 J. You need not calculate the exact sizes of the ellipses, but
How many tons of TNT are required for firing this gun? your diagram should show where the ellipses are larger or
(c) If the gun barrel is 500 m long, what must be the average smaller than the circle.)
acceleration of the projectile during firing? *61. (a) Since the Moon (our Moon) has no atmosphere, it is possi-
*56. An artificial satellite of 1300 kg made of aluminum is in a cir- ble to place an artificial satellite in a circular orbit that skims
cular orbit at a height of 100 km above the surface of the along the surface of the Moon (provided that the satellite
Earth. Atmospheric friction removes energy from the satellite does not hit any mountains!). Suppose that such a satellite
and causes it to spiral downward so that it ultimately crashes is to be launched from the surface of the Moon by means of
into the ground. a gun that shoots the satellite in a horizontal direction.
With what velocity must the satellite be shot out from the
(a) What is the initial orbital energy (gravitational plus
gun? How long does the satellite take to go once around the
kinetic) of the satellite? What is the final energy when
Moon?
the satellite comes to rest on the ground? What is the
energy change? (b) Suppose that a satellite is shot from the gun with a hori-
zontal velocity of 2.00 km/s. Make a rough sketch show-
(b) Suppose that all of this energy is absorbed in the form of
ing the Moon and the shape of the satellite’s orbit; indicate
heat by the material of the satellite. Is this enough heat to
the position of the gun on your sketch.