Can anyone give me the exact equations of motion for an object under gravitational force and their derivation?

You see, when an object, is falling, we usually take acceleration as constant because we take change in height negligible as compared to the earths radius. but if an asteroid is falling to earth we cannot do so. for every time, it keeps changing its distance from earth and hence changes its acceleration and thus its velocity. so taking into consideration, this variable acceleration, i want to get the equations.