We knew how to split atoms long before development on any atomic bomb began. We knew that splitting just one atom was powerful enough to make a single grain of sand move a bit. (incredibly impressive for just one atom.) The trick was how to get a ton of atoms to split and release their energy all at the exact same time.
Literally Einstein. e=mc^2
Essentially this means mass *is* energy. Like, super dense energy. Well, actually there isn’t much energy in one single atom (which has mass).
However, within say a single handful of a very dense element is an unfathomable amount of atoms.
From there you just have to ask if mass is energy, can you get it out somehow? Maybe splitting atoms apart? Let’s try that. Or maybe fusing them together, like the sun does?
Force balances, baby!
The nuclear force (very strong) holds the nucleus of atoms together. The electrical force (less strong) pushes all the protons in the nucleus apart. The two forces battle it out, but usually nuclear wins and the atom stays together.
At some point when the nucleus gets big enough, electrical forces (which act over a long distance) overpower the nuclear forces (which act only on short distances). Then you can sort of tap the nucleus (with a “slow neutron”) and the whole thing blows apart.
Basically: they understood force balances.
Source: I’m an applied mathematician and lots of physics based math comes down to force balances
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