Take two protons and set them 1 femtometer apart...

If you took two protons, stuck them together at the typical distance of an atomic nucleus, and there were no strong nuclear force holding them together, how fast would them be moving when they were a meter apart? That was the question I woke up pondering at 6 AM this morning. Being a bit groggy, and still half asleep, I tried using Coulomb's Law and Newton's Second Law with an integration, until I came to my senses and realized that the protons were not only moving relative to each other, but might possibly be moving relativistically, which would make the integration a whole lot more complicated. I did what I usually do when faced with a tough integration: look for an alternative way. Then, in a flash of inspiration, like the sun breaking through clouds, I found one: the Conservation of Energy! With that solved, I was able to go back to sleep (a good thing too, I need to catch up a bit over spring break).

Now, in the clear light of day, I decided to find out the answer, and I thought I'd share it with you, too.

Start with two protons 1 femtometer apart (1 femtometer = 1 quadrillionth of a meter [\(10^{-15}\)]). That's roughly the scale of an atomic nucleus. Using Coulomb's Law to find the force between the two protons, it turns out to be a whopping 230 Newtons! That's almost 52 pounds of force each single proton is feeling from the other! Which is absolutely incredible when you think about it...
Continuing on, the potential energy of each proton for this configuration turns out to be \(2.31\times10^{-13}\)  Joules. At a distance of 1 meter from each other, essentially all of their potential energy (>99.99.....%) will have been turned into kinetic energy, so using the Conservation of Energy and performing the calculations, I find that the protons are disappointingly classical, moving at a mere 8.1% of light-speed, or a little more than 24 thousand kilometers per second, or approximately 146 million miles per hour. At that speed their relativistic mass increase is a negligible 1.0067. Problem solved.

Still interested in the incredible force each proton feels, I calculated the acceleration they would be subject to at the instant of release and got an absolutely mind-blowing \(1.38\times10^{29}\) meters per second squared. That's... 10 billion billion billion times more than the acceleration we experience here on the earth's surface (10 billion billion billion g's, if you like).

The moral of the story? Be thankful for the Strong Nuclear Force today; its residual effects left over from holding quarks together inside of protons and neutrons hold those same protons and neutrons together in the nucleus.