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A5 9th Edition Solution

A5 9th Edition Solution - 12.101 PROBLEM 12.102 It was...

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PROBLEM 12.102 It was observed that the Galileo space craft reached the point of its trajectory closet to Io, a moon of the planet Jupiter, it was at a distance of 2820 km from the center of Io and had a velocity of 15km/s. Knowing that the mass of Io is 0.01496 times the mass of the earth, determine the eccentricity of the trajectory of the spacecraft as it approached Io. SOLUTION For earth, G M e = gR 2 = (9.81)(6.37 x 10 6 ) 2 = 398.06 x 10 12 m 3 /s 2 For Io, GM i = 0.01496 x GM e = 0.01496 x 398.06 x 10 12 m 3 /s 2 = 5.955 x 10 12 m 3 /s 2 h = r o v o = (2820 x 10 3 )(15 x 10 3 ) = 42.3x 10 9 m 2 /s ) 1 ( 1 2 0 H ± h GM r i 106 ) 10 955 . 5 )( 10 ) 10 ( 1 12 3 2 9 0 2 u u u ± i GM r h H 1 ² H 2800 km 2800 42 42 (2800 105.79 104.79 105.79 104.79 12.101
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7.307 x 10 6 m 7.307 k m 10.4 k m/s =10400 m/s = 75.96 x 10 9 m/s 2 13 2 6 10 24 . 14 ) 10 307 . 7 ( 3 8 m u u s 34 . 3749 10 96 . 75 ) 10 24 . 14 )( 2 ( 9 13 u u 1.0414 h 10.4 7307
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r A =1778.6 km =1.7786 x 10 6 m; r B = 3589 km =3.589 x 10 6 m 6 6 6 12 10 3676 . 5 ) 10 )( 10 1 )( 10 896 . 4 )( 2 ( u u u u = 3.412x10 9 m 2 /s 6 9 10 10 u u = 950.68 m 1.790 3.6 5.39 3.6 12.115 R=1737 km 3600 km 1790 km 1790 km 1.790 3600 km 3.6 3.422 3.422 B 950.56 m/s
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q ± ² 70 cos 5 . 1730 / 3590 1 5 . 1730 / 3589 = 0.444 ) 10 589 . 3 )( 556 . 0 )( 10 896 . 4 ( 6 12 u u = 3.126 x 10 9 m 2 /s 6 9 10 10 u u = 871 m/s =871-950.68 = -79.68 m/s 79.68 m/s 3600/1737 3600/1737 3.60 3.130 3.60 869-950.56 81.56 3.130 869.44 m/s 869.44-950.56=-81.12 m/s
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