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.

Assume the Earth to be a uniform sphere of mass M and radius R. Find the pressure P inside the sphere caused by the gravitational compression as a function of r, the distance from the center.

A meteorite of mass m collides with a satellite of mass 10 m. The satellite is orbiting a planet in a circular path of radius R. Due to the collision, the meteorite sticks to the satellite and the satellite goes into an orbit whose minimum distance from the planet is R/2. If the mass of the planet is M, what is the velocity of the meteorite before the collision?