Interestingly, it is possible in principle to get a good estimate of the total energy yield of the explosion from cell phone video of the event. The key is a fantastic example of dimensional analysis, a technique somehow more common in an engineering education than in a physics one. The fact that all of our physical quantities have to be defined by an internally consistent system of units is actually a powerful constraint that we can use in solving problems. For those interested in the details of this approach, you should start by reading about the Buckingham Pi Theorem. It seems abstract and its applications seem a bit like art, but it is enormously powerful.

The case at hand was analyzed by the British physicist G. I. Taylor, who was able to take still photographs in a magazine of the Trinity atomic bomb test and estimate the yield of the bomb. Assume that a large amount of energy \(E\) is deposited instantly in a tiny volume at time \(t=0\), and this produces a shock wave that expands spherically with some radius \(R(t)\) into the surrounding air of mass density \(\rho\). If you assume that this contains all the essential physics in the problem, then you can realize that the \(R\) must in general depend on \(t\), \(\rho\), and \(E\). Now, \(R\) has units of length (meters). The only way to combine \(t\), \(\rho\), and \(E\) into something with the units of

*length*is \( (E t^2/\rho)^{1/5}\). That implies that \( R = k (E t^2/\rho)^{1/5} \), where \(k\) is some dimensionless number, probably on the order of 1. If you cared about precision, you could go and do an experiment: detonate a known amount of dynamite on a tower and film the whole thing with a high speed camera, and you can experimentally determine \(k\). I believe that the constant is found to be close to 1.Flipping things around and solving, we fine \(E = R^5 \rho/t^2\). (A more detailed version of this derivation is here.)

This youtube video is the best one I could find in terms of showing a long-distance view of the explosion with some kind of background scenery for estimating the scale. Based on the "before" view and the skyline in the background, and a google maps satellite image of the area, I very crudely estimated the radius of the shockwave at about 300 m at \(t = 1\) second. Using 1.2 kg/m

^{3}for the density of air, that gives an estimated yield of about 3 trillion Joules, or the equivalent of around 0.72 kT of TNT. That's actually pretty consistent with the idea that there were 2750 tons of ammonium nitrate to start with, though it's probably fortuitous agreement - that radius to the fifth really can push the numbers around.Dimensional analysis and scaling are very powerful - it's why people are able to do studies in wind tunnels or flow tanks and properly predict what will happen to full-sized aircraft or ships, even without fully understanding the details of all sorts of turbulent fluid flow. Physicists should learn this stuff (and that's why I stuck it in my textbook.)

## 2 comments:

Your comment, "gives an estimated yield of about 3 trillion Joules, or the equivalent of around 0.72 kT of TNT. That's actually pretty consistent with the idea that there were 2750 tons of ammonium nitrate" begs the question, what is the conversion factor from ammonium nitrate to Joules or TNT?

Hi Mayer. According to wikipedia (which at least cites a paper in the literature), the "effectiveness" of ammonium nitrate relative to TNT is 0.32. So, 0.32*2.75 kT = 0.88 kT, which is not too far off from my very crude estimate.

There is a lot of uncertainty here. The efficiency of any ammonium nitrate explosion caused by fire will depend on a lot of variables (moisture content, how the stuff is stacked, just to name two). I now realize that using the visible cloud front (due to the pressure wave) as a proxy for the true shock position is not great. My distance scale estimates from the video + google maps are a guesstimate at best. Still, none of these factors individually is likely to blow the answer by an order of magnitude.

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