6.29 At point A on its earth orbit, the radius, speed, and flight path angle of a satellite are ra = 12, 756km, VA = 6.5992 km/s, and YA = 20°. At point B, at which the true anomaly is 150°, an impulsive maneuver causes Av = +0.75820km/s and Av, = 0. (a) What is the time of flight from A to B? (b) What is the rotation of the apse line as a result of this maneuver? {Ans.: (a) 2.045 h; (b) 43.39° counterclockwise}
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- Black holes are difficult to observewith telescopes because they, bydefinition, don’t emit or reflect any light. They can be found by look-ing for other nearby objects orbit-ing them, however. Here is a dia-gram of a star in a circular orbit around a black hole. a. The period of the star’s orbit is 90 days, and its orbital radius around the black hole isobserved to be 3.6 : ×10^11 m. Find the orbital velocity of the star in units of m/s. (You need to convert 90 days to seconds, first). The circumference of a circle is 2πr. b. The mass of the star is known to be 4 × 10^30 kg. Find the centripetal acceleration of thestar and the strength of the gravitational force on the star. c. Find the mass of the black hole.D Gm₁m₂ Fg KE = mv², Ug = - 2πr , ac = =²₁, v = ²7₁ T Gm₁m₂ GM g = G, Vesc = 2GM R , E = KE + Ug, G = 6.674 x 10-¹1 Nm²/kg² Problem 1: You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its radius to be 9 x 106 m and its rotation period to be 22.3 hours. You have previously determined that the planet orbits 2.2 x 10¹¹ m from its star with a period of 402 days (3.473 x 107 sec). Once on the surface you find that the free-fall acceleration is 12.2 m/sec². a) What is the mass of the planet? Answer: 1.5 x 1025 kg. b) What is the mass of the star? Answer: 5.2 x 1030 kg.Saturn orbits the Sun at an average distance of 1.43 × 10⁹ km with an orbital period of 29.46 yr. The Moon, which is one of the satellites of the Earth, orbits its parent at an average distance of 3.84 × 105 km with an orbital period of 0.07481 yr. (a) Use the above information to find the orbital speeds of Saturn around the Sun and of the Moon around the Earth. m/s m/s V Saturn VMoon = M = = (b) What is the expression for the mass M of the parent in terms of the orbital speed v of the satellite, the orbital radius R of the satellite and the gravitational constant G? (Do not substitute numerical values; use variables only.) (c) Now use your answers from parts (a) and (b) to find the ratio of the mass of the Earth to that of the Sun. ME Ms
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