How to expand this equation considering acceleration due to gravity into 3D vector space?
How can we expand this following equation into 3D vector space? I learned this equation from this answer: Don't heavier objects actually fall faster because they exert their own gravity?
The answer shows how we can find time as function of distance (or single dimension) when there's force due to gravity accelerating both pieces of mass as follows:
Where and are the masses of two bodies, is initial distance from mass to another and is the final distance from mass to another.
If this equation had just the force caused by gravity within it, I could easily divide it into components, and use this equation for each dimension, but as seen, this equation only has masses and distances in it.
How can I apply this kind of equation into 3d vector space, ie. get the position as 3d coordinates as function of time when initial position and velocity are known?
If this equation cannot be applied in to 3d space, then how could we derive another equation, which applies same kind of relations into 3d vector space? (I checked the answer I linked above, but it goes somewhat above my understanding, as differential equations are used.)