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: t = 1 2 G ( m 1 + m 2 ) ( r i r f ( r i − r f ) + r i 3 / 2 cos − 1 ⁡ r f r i ) Where m 1 and m 2 are the masses of two bodies, r i is initial distance from mass to another and r f 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.)

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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&#039;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&#039;s force due to gravity accelerating both pieces of mass as follows:  t  =      1          2      G      (              m        1            +              m        2            )                  (                  r      i              r      f        (          r      i        −          r      f        )    +      r    i          3              /            2            cos          −      1        ⁡                    r        f                    r        i                        )      Where       m    1   and       m    2   are the masses of two bodies,       r    i   is initial distance from mass to another and       r    f   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.)

Eden Bradshaw · Accepted Answer

Two issues:1) the equation was derived assuming that the initial speeds were zero, so both masses started at rest. To get a more general expresion yuo need to integrate again (I&#039;ll check later if this is easily doable)2) you can apply this equation in 3d, the equatiosn assumes that the masses start at rest and follow a staight path until they collide. in 3D you just need the initial positions of the masses in space, and calculate the initial distances. both r&#039;s will be along the line that pass trhrough the masses.

How to expand this equation considering acceleration due to gravity into 3D vector space? How can w

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