# How One Man Derived an Equation of Atomic Bomb Blast Radius by Just Looking at the Photographs?

When assessing the technical implications of dropping a nuclear bomb (or other weapons with explosive charges), notably one aspect of the core of experiments and research conducted in the stages leading up to the ultimate dropping. One key aspect to consider was the bomb blast radius, or in other words, an assessment of how far damages would spread outwards from the epicenter of the nuclear reaction and how their severity would decrease with that distance. This would then have several strategic implications, such as adapting the explosive load’s mass or assessing where to drop the bomb.

## Theoretical basis

The formula for the bomb blast radius was initially conceived through observation. A mathematician, G. I. Taylor, from the British delegation, had been sent to participate in the first atomic bomb trials in the Manhattan Project and witnessed the Trinity test (the first detonation in history). Using scaling analysis and photographs taken at the test site, Taylor worked out the blast radius and eventually publicly disclosed an atomic bomb’s inner workings.

The scaling analysis process, otherwise known as dimensional analysis, relied on Taylor outlining the important factors and variables – as well as their units – which might affect the blast radius of a bomb, its rate of propagation. He identified energy and time as the first two parameters, and there remained a property of the surrounding air, which had to be taken into account. Taylor ruled out the pressure as the bomb’s pressure was significantly larger than atmospheric pressure and would thus not be influenced by it. He concluded that it would thus either be air density or temperature, and he chose air density.

Therefore, he needed to combine energy, time, and air density, using their units, to find a result only dependent upon a distance in meters. Energy (in joules, or kilograms times meters squared per seconds squared), time (in seconds), and air density (in kilograms per meters cubed) all combined into one neat expression and concluded that the blast radius of a bomb depended upon energy to the power of 1/5, time to the power of 2/5, and air density to the power of -1/5.

However, this method did pose some limitations for dimensional analysis, only allowed him to determine which parameters were important. They still could and needed to be scaled by a constant, which turned out to vary in the type of bomb or explosive used.

## Practical implications

The formula now needed to be tested or put into practice again. In Taylor’s case, he tested his formula against the photographs he had taken at the test site at different time intervals and validated his hypothesis. He worked out an experimental constant to scale his formula to fit the observed radius.

However, when it comes down to dropping the bomb, assessing these variables becomes slightly more complicated. While the energy may be determined beforehand, the air pressure may greatly fluctuate depending on weather and might not be predicted as easily as the other variables. This becomes problematic when the bomb must remain contained within a certain radius, and conditions must be assessed at the site to make the most precise approximation and prediction.

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