How do I calculate the actual gravitational potential energy between two objects $m_1$ $m_2$ that are at zero distance?

Question

How do I calculate the gravitational potential energy between two objects that are separated by near to infinite distance? I would like to know if the two objects are kept at near infinite distance, then at what kinetic energies would they come and collide into each other.

The way to calculate gravitational potential difference is $$U = -G m_1 m_2 (1/r - 1/R).$$ Where in I can enter R = infinity. However, when I put r = 0 it gives me infinite energy, which doesn't seem possible. That would mean r will have a positive value. What is the value?

(This question is out of curiosity and not for any assignment.)

Answer 1

I would like to know if the two objects are kept at near infinite distance, then at what kinetic energies would they come and collide into each other.

When two real objects collide they won't collide at a distance of $r=0$. They'll collide at a non-zero distance because they have non-zero size.

Once they collide (touch) they will cease to obey that convenient form of the gravitational potential because they'll be be one complex body that doesn't have such a neat form of potential energy.

So the basic premise is wrong : they never reach $r=0$ as two distinct bodies.

$r=0$ would only apply to point masses and a point mass would in any case have an infinite density and all sorts of problems arise when you start plugging infinite numbers in.