Bomb Blast Radius - Numberphile

Key Takeaways

we're going to calculate the blast radius of an atomic bomb fortunately without dropping any atomic bombs because that's one way to do it and that was exactly how the u.s military were doing this 1940s manhattan project los alamos all of that part of history interesting things happening in the physics and maths world the us military are doing all these experiments and we're making measurements of as they change the the size of the bomb or certain weather conditions as to how fast that sort of mushroom cloud would expand of course quite useful information if you're dropping a bomb you know potentially near your own soldiers or there are certain areas you want to avoid you want to have an idea how much that's going to spread out pretty important information and as because atomic bombs were so new back in the 1940s we didn't really know this so it did require this like trial and error approach or it required the clever mathematical approach and so this is a story of g.i taylor a british mathematician who was asked by the uk government to help out during the world wars so he actually helped out in world war one and world war ii in um world war ii he was sent over as part of the british delegation to go and visit the test site to visit the manhattan project and he actually saw the first ever blast the trinity test in 1945 he actually witnessed one of i think there are only 10 people that were there and one of them was this mathematician gi taylor what he did using some photographs taken from the explosion was actually calculate the formula for the blast radius of the atomic bomb which at the time was a u.s military secret so d.i taylor did these calculations i just published this in mathematical research journals and it was quoted as pretty embarrassing for the u.s military at the time because this was like their most closely guarded military secret about the inner workings of an atomic bomb and then this british mathematician comes along and just kind of you know does a pretty straightforward calculation as we'll see to just get the answers basically so it's a very very cool story and also uses possibly my favorite tool in all of applied maths which is that of scaling analysis it's very very applied math should stress this because you kind of go into physics and think about what is happening in a problem to then write down the important parameters in your problem and then combine them in such a way to get the answer you want so you see you do have to have that sort of very applied physicist like mind to be able to use this but it's such a good tool because you can calculate the formula for an atomic bomb without setting off a single atomic bomb and so the way taylor did it first of all what factors are going to influence the spreading rate of our bomb blast first thing at least that came to my mind and to taylor's mind was the energy you have a bigger bomb it's gonna blow up more stuff it's reasonably straightforward to see so we don't know whether that relationship is linear or anything yet but we know that the energy of the bomb is going to affect that blast radius so we're going to write down energy we're going to label e that's definitely one of the parameters that's going to be in our problem now the other one that's reasonably obvious is time if you've just set your bum off it's not gone very far if you wait a few seconds it's gone a lot further we're trying to work out the radius of the blast so as time increases the radius will certainly increase so we're going to have time which we label with t and now the sort of real fluid dynamics and physics comes into it and taylor was a fluid dynamicist in cambridge and this is where this really helped him because he thought about the problem as follows you have as the bomb detonates you have a huge amount of energy being released from a tiny tiny volume and so you have a very almost instantaneous explosion from tiny tiny volume to massive amount of energy and what that's going to do is create a massive massively high pressure within that explosion that sort of cloud of your bum so you can kind of think of it as if this is the sort of initial explosion in here if you take this to be pressure this is really high inside this region and as soon as you get out of the bomb region the atmospheric pressure is really low and you almost have like a what we would call a shock this is where the term shock wave comes from the shock wave of a blast because here you've got super high pressure from this giant explosion and here you've just got atmospheric pressure and these are like orders of magnitude different what this tells you is not what is important in the problem but what is not important and this is the sort of the fluid dynamics the physical intuition of these problems and taylor was a fluid dynamicist why he knew this so because this is so high and this is so small the pressure of the air is not going to play a role in the expansion of the bomb so whether it's high pressure or low pressure weather system it's not actually going to matter so it tells you that pressure is crossed off if the air pressure isn't important then you think well what other properties of the air are going to affect the spread of the bomb you would expect there to be a difference in the spreading rate depending on the fluid or the air that you're moving through the other possibilities are density and maybe temperature we have an equation relating these two it's called the ideal gas law and you often will model air as such and it says that these two if the pressure is constant or negligible these two just depend on each other so he just picked density so this sort of physical intuition told you pressure didn't matter and that meant it had to be density that was the key insight that came from taylor being a fluid dynamicist the pressure of the this atmosphere doesn't matter but doesn't this pressure matter this pressure will absolutely matter but this will basically be controlled by the energy this is the the thing with scaling analysis and it's a very good question because there are so many different things you can pick and sometimes it is almost like black magic that you pick the right ones and put them together in such a way so i'm sure there are many many examples of scaling analysis not working but this just happens to be like possibly my favorite example of it working and discovering a military secret so we've got energy we've got time and then we've got density density of the outside air which we label with rho the scaling analysis is what comes in here because what you do is you look at the units of your three parameters we denote this with a square bracket in maths so let's start with time because that's easy the units of time are time seconds i'm going to just call it a time which is a t now energy the way i remember the units of energy is to think of let's take my favorite energy kinetic energy so kinetic energy is a half times mass times velocity squared so the units are a mass times a velocity squared velocity is distance divided by time so that's distance over time squared so i now have a mass a length squared over a time squared for the units for energy i've got just a t for time and density is a mass per length cubed because we're working in 3d kilograms per meter cubed for example so we've now got the units of energy the units of time and the units of density and now the scaling analysis argument is to say well i have determined through knowledge of physics and fluid dynamics as taylor did that these three are the most important parameters that i think control the blast radius so if i can arrange these in such a way that i get rid of the mass and get rid of time from the units and all i'm left with is a length that must be the radius because what other possible length could it be that's the kind of argument and then of course you have to validate this and test this with some amount of experiments so you don't need to do hundreds like the us military but taylor happened to have one or two photographs from that one test he went to and he checked it against that and showed that it was correct so what we just have to do now is combine the three sets of units in such a way that we just get length we've got a mass a length and a time so i think if we start by removing mass so if we do energy e divided by density rho then that is equal to square brackets for units m l squared over t squared divided by the density puts an m on the bottom and an l cubed on top so the m's cancel and so this is equal to l to the 5 over t squared so now we've got rid of mass which is very good but we still have time so we get rid of time by introducing t so we just are going to have to multiply on the top by time squared and that gives us now with the time squared on the top we just have l to the five because it will cancel and finally this is a length to the power five but we just want a radius which is just l so we just take the whole thing to the power of one-fifth so if we want our time-dependent radius of an atomic bomb the radius of an atomic bomb is proportional to the energy to the one-fifth the time after explosion to the two-fifths and the air density to the minus one-fifth and that right there was the most closely guarded secret in u.s military in the 1940s had the u.s military figured it out with mathematics or had they just used a number of tests and sort of you know guessed roughly so so if you were doing this experimentally which is what they did you would do lots of different explosions and you would take measurements at different times as the thing is exploding and expanding you'd make take do it on different days with different air densities and you'd use different sized bombs and you'd have all these data points and then you do sort of try and find a scaling that makes the data collapse into a straight line that's generally how you do applied maths research today that's how i did my experiments doing my research in fact so you just get lots of data points and try and plug it all together and then sort of it's kind of trial and error and then suddenly it it perfectly fits this pattern and taylor was able to check he got it right because he had a few pictures from the trinity test yes he did he had these two pictures he actually published them in a magazine and when he released this this paper he published these two images um and so he using um he could check this against against these actual images to check the radius and of course what we are missing here is there's going to be a constant here so this is proportional to so the units match up but you can still have a number in this gap and now that's the kind of thing you need to do experiments to get and taylor only had the one to compare to and so i think he was trying to estimate the size of the bomb dropped in this trinity test in 1945 and he believed it was 20 kilotons sorry he believed it was 22 kilotons and the actual size was 20 which given the this that gets pretty good let's be honest given that you know he didn't have access to highly classified us military secrets he was able to get within a 10 accuracy the size of the bomb that was dropped in that first ever nuclear test this isn't going to help you build an atomic bomb this is just going to give you an idea how devastating they could be absolutely so this this would be one of the first things given some new some new weapon or some new even sort of like a new source of energy one of the first things you would want to do would be to try it at different values of energy different sizes different densities and see what happens sort of before you were able to actually use it safely well i guess a bomb's not really safe in a controlled way in a controlled way thank you without unintended circumstances you would need to know this kind of information so this was as i said this was literally done based on the first ever detonation of an atomic bomb is this constant this unknown number here dependent on the type of bomb or would a tnt be different to an atomic bomb or i believe so that's the sort of the limitation of the scaling analysis whilst you can get which parameters are important and the powers of those parameters you can never get that constant but once you've got the constant it works for all atomic bombs yeah there's going to be some error absolutely going to be some error but yes once you have that constant it hopefully would hold for for a particular at least a particular type of atomic bomb you had the same bomb and you dropped it and then you doubled your energy this would predict how much more the radius would be and you would absolutely expect that to follow what does it tell us well time is certainly increasing the most in the sense of it has the highest power if time doubles versus energy double or density doubling you're going to see that the biggest change being in time but what it just tells us as you increase the energy the the radius will increase but it's also telling you here if you were to go from e to e squared you're only going to get quite a small increase in your radius so this one fifth is sort of saying that you really have to ramp up the energy to get that large radius where you know compared to this being a larger power because it's quite a small power and because the density is negative that's on the denominator so as the density gets higher the radius decreases and that's always a good sanity check for these things making sense if the energy goes up you expect a bigger bomb if the you wait longer you expect a bigger radius and if you increase the density of the air so you imagine there's more water in the air it's like thicker it's harder for the explosion to move through thicker air so you would expect it to shrink so there's always that nice as i said the sanity check that it does make physical sense and and mathematically the unit's balance the scaling works and then you would then need to do some extra at least one experiment to get a grasp on that number but for the us the way you would think about it is the the density of the air is reasonably constant say you're dropping the bomb in new mexico the air density in the summer will be approximately the same across the summer period so you know that the time that just will constantly increase so the question would then be say you wanted a bomb such that within two seconds of it being dropped it had destroyed a one kilometer radius area you then plug in one kilometer as your radius you say i want within two seconds it to be gone i plug in my air density it tells you the size of the bomb you have to drop to do that damage so so that's this really is quite valuable information if you're trying to do like a controlled sort of targeted bomb we're releasing energy and it produces two neutrons each of which go on to react or to split other atoms of plutonium so you get a chain reaction and the explosion just builds up and up and up