Two Theoretical Bodies in Space, Time to Collide.

[Zemy]

Firstly, I am new here, sorry if this was posted to the wrong thread.

Upon seeing a statistic online that "if two baseballs were put in an empty vacuum only a meter apart they would collide in 3 days due to their own gravitational effect on each-other." I set out to firstly prove this was true and secondly to see if there was a relationship between this and something with the same radius, and distance from each other. 

So by using the equation G*[(m₁*m₂)/r²] =a   I made a range of values for acceleration by varying the radius from 1 to 0.1 in graduations of -0.1 . Using the mass of a baseball as 0.145 Kg (therefore m₁ & m₂ = 0.145) and taking G as its known value I made all values of a from 1 meter apart to 0.1 meters apart. I then calculated the mean average from these results so i did not have to deal with a changing acceleration in my calculations. This average came out to be 2.17467*10¹¹

Now I was trying to calculate the time taken for this to happen, for this I used the SUVAT equation s = ut + 0.5at² rearranged to give t = (2s / a)^0.5 and when I used the average acceleration and the distance between the surfaces of the baseballs (taking a baseballs radius to be 0.0365, so therefore s = 1 - (2 * 0.365)) it gave 291983.1918 seconds which is about 3.37 days which is an acceptable value for 3 days, if the source had rounded it to the nearest day.

I then repeated this process with the same radius and distance but with different masses and therefore accelerations.  I tried it with the mass of our sun and a control with a mass of 1.

After doing this and working out the subsequent times for each I wondered if the mass of the object was inversely proportional to the time taken  for them to collide. After doing time multiplied by mass to work out K I noticed it was the same for all three calculations I had done. The value was 42337.56 . Does this mean that under conditions where the radius of an object is 0.0365 m and the distance between the two objects is 1 m all you have to do to work out time is multiply the mass by 42337.56? Although this is only at the boundary of the two objects rather than their centers of mass. However this is practically stating that m*t = 42337.56 when r = 1 and s = 1 - (2 * 0.365).

Or m*(2*s / {G*[(m₁*m₂)/r²] })^0.5 = 42337.56

Feel free to correct me if I'm being an idiot.

Many thanks.

Edited December 13, 2018 by Zemy

[Janus]

There's an equation that will give you the exact answer.  it is

[latex] T = \frac{\cos ^{-1}  \sqrt{\frac{x}{r}}+\sqrt {\frac{x}{r} \left ( 1- \frac{x}{r} \right )} }{ \sqrt {2G (m_1+m_2)}} r^{\frac{3}{2}}[/latex]

Here r is the distance between the center of the balls at the start and x the distance between the centers when they collide (2 times the radius of the balls themselves)

Using this method, I get an answer of  ~2.9 days.

 

Edited December 14, 2018 by Janus
Latex fix

[Zemy]

Cheers, I did not know of this equation, how is it derived?

[Janus]

You have to use integral calculus to account for the fact that the force, and thus the acceleration, changes as the distance decreases.