Question

An astronaut is in equilibrium when he is positioned distance D from planet X and 581 km from planet Y, along the straight line joining the planets' centers. Let Mx and My be the masses of planets X and Y respectively. If the ratio Mx/My is equal to 0.0581, the distance D (in km) between the astronaut and planet X is: I know the answer is 140 km but I don't know why. Please help!

Answer 1

\[Fx = GMx*m/D^2\]
\[Fy = GMy*m/Dy^2\] where Dy = 581km

for equilibrium Fx = Fy
hence,
\[Fy/Fx = My/Dy^2 * D^2/Mx = 1\]
\[D^2/Dy^2 = Mx/My\]
\[D^2 = 581^2 * 0.0581\]
hence D = 140.04 Km