system
Given the following post, respond as if you're commenting on Reddit:
human
During the first nuclear bomb test, how did they know how far away to observe the blast? Did they already know how many tons of TNT (or TJ) it was?

I tried some google-fu, but nothing showed up. How did the scientists that made the bomb know where to place their observation cameras and bunkers? From what I've seen they were fairly close enough to the blast to see it without using binoculars or other things. There were bunkers 10,000 yards away and observation things 20 miles away. 

I also read that there was a betting pool as to how many tons of TNT or TJ the blast would be.

Is there a way to figure this out mathematically? (the TJ and distance of the blast wave)
gpt
The question of how large a nuclear blast can be is a function of the total energy release possible multiplied by the expected efficiency of the weapon. 

So in the case of the first nuclear weapon, they knew that it contained 6.2 kg of plutonium. 1 kg of plutonium, completely fissioned, releases 18 kt or so of energy. So the absolute upper bound is around 112 kilotons. 

But they also knew their method was not going to be very efficient. The "safe" estimate was that it would release about 5 kilotons worth of energy — an efficiency of only about 5% (that is, only about .28 kg of plutonium would fission, in a weapon that had 6.2 kg of plutonium in it). The most optimistic estimates (e.g. in the betting pool) was around 50 kilotons — an efficiency of 45%. But this was unusual — most thought it would be lower, maybe even less than a kiloton. 

In reality, it was around 20 kilotons, which is an efficiency of about 18%, which is considerably higher than the "safe" guess (4X more). 

They had a pretty OK idea of how the blast wave would react at different distances and kept anything really sensitive well outside of those distances. They were aided in the fact that damage does not scale linearly with increase of explosive yield: because the explosive energy is released as a sphere, it scales as a cubic root. To put this simply, in order to double the area of destruction, you have to increase the yield by a factor of eight (so a 8 kt bomb destroys twice the area of a 1 kt bomb). So just because it was 4X more energetic than they predicted, that did not mean they had to be 4X as far away to be safe. 

There were many uncertainties, however. They tried to keep things as safe as they could under the circumstances, but there were many unknowns. They kept all personnel at comfortably safe distances, and tracked the spread of radioactive fallout. They even prepared for grim eventualities such as people accidentally dying during the test (which sounds fanciful, but it is worth noting that during several Soviet nuclear weapons tests there were inadvertent deaths due to unusual unanticipated phenomena, such as reflection of blast waves off of inversion layers in the atmosphere).