Collisions of large bodies starting at stationary positions

[Silverious]

I got up this morning, and decided I was going to try to calculate this. So here is what I asked.

Given 2 stationary bodies, starting some distance apart, of different mass, where would they collide in respect to one another?

I tried using Newton's gravity formula to calculate the force of attraction every few intervals of distance, and then graph the acceleration. But I figured, if I could find out how to calculate the acceleration of acceleration, it would make my calculations much simpler. (As in, as they accelerate toward each other, the force increases, so they accelerate even greater.)

I was wondering if anyone could help me with this.


Btw, this is my first post in these forums, so I hope this is appropriate. Excuse me if it isn't.

 

[enigma]

Hi Silverious,

welcome to the forums!

Do you know calculus and differential equations?

If you do, the problem becomes more or less trivial.

 

[Janus]

If you are just interested in where compared to their respective starting positions the two bodies will collide, that is easy, they will collide at their barycenter. (their Common center of gravity).

Two find the barycenter, you can use the formula:

[tex] D_{2}= D_{1}\frac{M_{1}}{M_{1}+M_{2}}[/tex]

This will give the distance from M₁ to the barycenter.

D₁ is the initial distance between the two masses

 

[Silverious]

Wow, thanks. I feel kind of foolish. But thanks anyways.

 

[Ambitwistor]

As Janus said, if momentum is to be conserved, then the center-of-mass of the system has to remain stationary, so that's where they'll end up. If you don't like that argument, you could integrate the differential equation of their motion directly, but it might be difficult; I seem to recall that when I solved this problem for the amount of time it takes for them to collide, I got an elliptic integral, so distance as a function of time would have to be the inverse of an elliptic integral.