What should the radius of the orbit be (measured from the center of Mars), if we want the satellite to take 8 times longer to complete one full revolution of its orbit

Answer and Explanation:

For a satellite of mass m, orbiting at a distance r from the center centrpetal force = gravitationa pull:

GmM/r2 =mw2r

or GM =w2r3 = (2Pi/T)2 * r3

or (T1/T2)2 = (r1/r2)3

Given TT2/T1 = 8, r1 = 2.2rmars + rmars = 3.2rmars

(r2/r1)3 = 8 2 = 64 = 43

(r2/r1) = 4

r2 = 4 * r1 = 4 *3.2rmars = 12.8rmars

So distance of  the satellite from the center of  Mars = 12.8rmars = 12.8 * 3.397 x 106 m  = 4.348 x 107 m

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are you referring to matrix mathematics!