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

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.)

• $r$ must be a distance and distance are always positive (so $r=0$ has no pyhsical sense, since it'll mean that both objects are on the exact same place). – MatMorPau22 Oct 30 '17 at 15:54
• – Qmechanic Oct 30 '17 at 15:59
• Ahh.. The $r=0$ singularity in Newtonian gravity. :) – Dr. Ikjyot Singh Kohli Oct 30 '17 at 16:03
• If real objects collide they must have some size. How far apart are the centers of mass when the surfaces touch? for example, two spheres would be $r=r_1+r_2$ apart when they collide. – Bill N Oct 30 '17 at 16:03
• @BillN consider two atoms. Or say even two quark type particles – rahulg Nov 1 '17 at 10:13

## 1 Answer

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.

• consider two atoms. Or say even two quark type particles – rahulg Nov 1 '17 at 10:14
• We do not have a complete theory of quantum gravity so we do not have a way to answer that properly. There is Does gravity exist between quarks in a neutron ? which will give you some insight into this, but also note that the Heisenberg Uncertainty Principle makes the image of point-like masses meaningless. Gravity is also a very weak force compared to the other interactions that such particles experience, so it's not really an issue. – StephenG Nov 1 '17 at 18:25