# Rare gravitational problem

1. Dec 4, 2011

### kosmologo

Somebody asks me a problem of a gravity that I could not solve so I ask for help of teachers or experts. The problem is as follows:
In space we have 2 very special bodies. The first one is located at a point A and the second on point B.. The two points are separated by a distance "d". Point A is the center of a sphere of radius "r" variable in time but constant density ρ. The variation of the radius "r" is exponential with time, where K is a constant and t is time.

r = e^(Kt)

Therefore its mass M is also a variable of the form:

M = 4 / 3 * ∏ * ρ * r ^3

On point "B" is a particle of unit mass which has the characteristic of being gravitationally repelled by the sphere, as if the sphere were of normal matter and the particle of antimatter. This particle is located initially at t = 0, at distance "d" from the center of the sphere and at a distance X, after an elapsed time "t" any different from zero. The repulsive force increases following the same change in the law of universal gravitation, but obviously with the opposite sign.
The problem require to determine the value of X as a function of time.
Eternally grateful if you give me the solution.