Question

Satellite X, a small moon of Planet X, has an orbital period of 1.77 days and an orbital radius of 4.22 x 10^5 km.

a) What is the mass of Planet X?

b) Assume that Satellite X moves in a circular orbit. What is its orbital speed?

c) A small mass of m = 2.5 kg is dropped from rest from the moon. What is its speed when it reaches the surface of Planet X, whose radius is 71,492 km?

d) What is the escape speed from Planet X?

Answer 1

a)

using kepler's 3rd law of planetary motion

T^2 = 4 pi^2 R^3 / (GM)

(1.77* 24* 3600)^2 = 4* 3.14^2* (4.22* 10^8)^3 / ( 6.67* 10^-11* M)

M = 1.9* 10^27 kg

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b)

v^2 = 2 pi * 4.22* 10^8 / ( 1.77* 24* 3600)

v = 17329.5 m/s

c)

using conservation of energy

0.5 m v^2 = GMm ( 1/ r - 1/ ( r+h))

0.5 v^2 = 6.67* 10^-11* 1.9* 10^27* ( 1/ (71492* 10^6) - 1 / (4.22* 10^8))

v = 54265 m/s

d)

v^2 = 2 GM / r

v = 59542.35 m/s

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