Miranda, a satellite of Uranus, is shown in part a of the figure below. It can b
ID: 2198485 • Letter: M
Question
Miranda, a satellite of Uranus, is shown in part a of the figure below. It can be modeled as a sphere of radius 242 km and mass 6.68 1019 kg. (a) Find the free-fall acceleration on its surface. __________________m/s2 (b) A cliff on Miranda is 5.00 km high. It appears on the limb at the 11 o'clock position in part a of the figure above and is magnified in part b of the figure above. A devotee of extreme sports runs horizontally off the top of the cliff at 7.20 m/s. For what time interval is he in flight? _____________________s (c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? _____________________m (d) What is his vector impact velocity? ________________m/s __________________Explanation / Answer
(a) Surface gravity a = [GM] / r^2 a = Acceleration G = Universal Gravitational Constant M = Mass of Miranda r = Radial distance from center Given: a = ? G = 6.67428E-11 m^3/kg-s^2 M = 6.68E+19 kg r = 242,000 m Solve a = [ (6.67428E-11 m^3/kg-s^2) * (6.68E+19 kg) ] / (242,000 m)^2 a = [ 4,458,419,040 m^3/s^2 ] / (58,564,000,000 m^2) a = 0.0761 m/s^2 (b) Time of flight of crazy sports fan... if there's an angle, I don't see it, so horizontal launch H = 5,000 m Vo = 7.2 m/s g = 0.0761 m/s^2 (as above) t = SQRT { [2H] / g } t = SQRT { [ 2 * (5,000 m) ] / (0.0761 m/s^2) } t = SQRT { [ 10,000 m ] / (0.0761 m/s^2) } t = SQRT { 131356 s^2 } t = 362 s (c) Distance from cliff, assuming a perfectly flat landing zone Vo = 7.2 m/s t = 362 s R = Vo * t R = (7.2 m/s) * (362 s) R = 2606.4 m (d) Impact velocity Vy = SQRT { 2gH } Vy = SQRT { 2 * (0.0761 m/s^2) * (5,000 m) } Vy = SQRT { 761 m^2/s^2 } Vy = 27.6 m/s