Question
Please help, i posted this question earlier but no one was able to solve it, pleassee :/ ill rate lifesaver to anyone who saves my life with this problem. I am really stuck here.
On December 25, 2004, the Huygens probe separated from the Cassini spacecraft orbiting Saturn and began a 22 day journey to Saturn's giant moon Titan (see the figure below ), on whose surface it landed. It is useful to know that Titan is 1.22×106 km from the center of Saturn and has a mass of 1.35×1023 kg and a diameter of 5150 km .
At what distance from Titan should the gravitational pull of Titan just balance the gravitational pull of Saturn?
Explanation / Answer
The radius of the Titan is R = 5150 km/2 = 2575 km The distance of Titan from the center of the Saturn is d = 1.22*106 km The mass of Titan is M = 1.35*1023 kg Let at a distance r from the surface of Titan, the gravitaional pull just balances that of the Saturn According to Newton's universal gravitational law, the gravitational pull on the satellite (mass be m) due to Titan at this position is F = GMm/r2 Where G = 6.67*10-11 Nm2/kg2 is the universal gravitational constant At this position the satellite will be at a distanec d-r from the Saturn F = GM'm/(d-r)2 Where M' = 5.68*1026 kg is the mass of Saturn For the gravitational pull to be balanced, GMm/r2 = GM'm/(d-r)2 M/r2 = M'/(d-r)2 (M'-m)r2 + 2Mdr - Md2 = 0 5.68*1026 r2 + 3.294*1032 r - 2.00934*1041 = 0 r = 1.85*107 m r = 1.85*104 km M/r2 = M'/(d-r)2 (M'-m)r2 + 2Mdr - Md2 = 0 5.68*1026 r2 + 3.294*1032 r - 2.00934*1041 = 0 r = 1.85*107 m r = 1.85*104 km (M'-m)r2 + 2Mdr - Md2 = 0 5.68*1026 r2 + 3.294*1032 r - 2.00934*1041 = 0 r = 1.85*107 m r = 1.85*104 km