The space shuttle is releasing a 1000-kg satellite into a stable circular orbit

Question

The space shuttle is releasing a 1000-kg satellite into a stable circular orbit at an altitude of 500km above the earth. (a) If "releasing" the satellite means simply letting it go (no extra boost), how fast does the shuttle need to be traveling when it releases the satellite? (Take the radius of the earth to be R_E = 6, 370 km, and its mass M_E = 5.97 times 10^24 kg.) (b) What is the angular momentum of the satellite in this orbit? (c) What is the gravitational potential energy of the earth-satellite system? Based on this result, what would be the escape speed from the earth for a satellite in this orbit? (d) The sun has a mass of approximately 2 times 10^30 kg and is about 1.5 times 10^8 km away from the Earth. What is the gravitational potential energy of the sun-satellite system? Based on this result, what would be the escape speed from the earth-sun system for a satellite in this orbit?

Explanation / Answer

(A) G M m / r^2 = m v^2 / r

v^2 = G M / r

v^2 = (6.67 x 10^-11) (5.97 x 10^24) / ( 6370000 + 500000)

v = 7613.3 m/s

(b) L = m v r

= 1000 x 7613.3 x (6370000 + 500000)

= 5.23 x 10^13 kg m^2 / s

(c) PE = - G M m / r

= (- m) (G M / r)

= - m v^2

= - 1000 x 7613.3^2

= - 5.80 x 10^10 J

- m v^2 + m v_eascape^2 /2 = 0

v_escape = sqrt(2) v = 10766.8 m/s

(d) PE = - G M m / r

= - (6.67 x 10^-11) (2 x 10^30) (1000) / (1.5 x 10^11)

= - 8.89 x 10^11 J

1000 v^2 /2 = 8.89 x 10^11

v = 42174.24 m/s

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