A planet is moving around the sun (mass M_s) in an elliptical orbit. When it is at a distance r_o from the sun, its velocity is v_o and the angle between the radius vecor r_o and the velocity vector v_o is equal to ø. Find the maximum and the minimum distance that will separate this planet from the sun.

An artificial satellite (mass m) of a planet (mass M) revolves in a circular orbit whose radius is n times the radius R of the planet. The satellite experiences a slight resistance due to cosmic dust. The resistance force is dependent on the satellite velocity as F = a v², where a is a constant. Calculate how long the satellite will remain in orbit before it falls on to the planet's surface.

A particle of mass m is located on the outside of a uniform sphere of mass M at a distance r from its center. Find the potential energy of gravitational interaction between the particle and the sphere.