Gravitational attraction between two kilogram weights [duplicate]

Question

Imagine two weights, each of mass one kilogram, floating in outer space. Starting one meter apart, and under no forces other than their own gravity, how long would it take before these two weights collided? To simplify the problem, assume the weights are actually point masses each of mass 1 kilogram.

So far, I've started with the fact that each weight feels a force of gravity equal to G/R^2, as each of the masses is 1 kilogram. However, this force changes as the weights keep getting closer because the R value would decrease. Where could I go from here? Feel free to use calculus as I am familiar.

Answer 1

You have discovered differential equations. Step one is write an equation capturing the change in $r$ vs $r$. This is done with Newton laws (the acceleration depends on one over distance squared):

$$ \ddot r(t) = -\alpha/r(t)^2$$

where $\alpha$ captures all the constants, and $ \ddot r $ is the second time derivative of $ r $.

The next step is to solve the differential equation generally. You need to find a function of time that works. One way is to guess a form and plug it in:

$$ r(t) = at^b $$

Then plug that in using:

$$ \ddot r(t) = ab(b-1)t^{b-2} $$

and

$$ -\alpha/r(t)^2 = \frac{-\alpha}{a^2}t^{2b} $$

Then solve algebraically for $b$, which is $2/3$.

So the cube of the radius goes like time-squared? Sounds like Kepler's Law. Maybe you could have solved the problem with dimensional analysis.

Anyway, from there: make that function fit your problem.