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Cook's Edit to Taylor's Formula

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Introduction
I'm proposing a new formula used for calculating high speed, high yield explosions. The formula is an edit to Taylor's formula, which was a dimensional analysis used to determine the yield of the atom bomb; however, Cook took released government data and extensive lab testing to fix Taylor's, yielding an accuracy of roughly 2% error.

Cook's Formula
You can read all about the derivation of the formula here, but I will also link the relevant formula here. Where gamma is generally 1.4 for standard ground explosions, rho is the density of air, R is the radius of the fireball formed in a time t, and t is the time it took the fireball to get to that radius.

Proposal
A good exert from Cable (found at the bottom of this page) on why it is valid:

“But wait!“, I hear you cry. Isn’t all of this used to determine the power of a nuke anyway? Why yes, yes it is. But it SHOULD apply to any explosion and here’s why: The time factor is what, as I mentioned and showed above, separates nuclear, high, and low explosives. How fast a bomb releases its’ energy can actually change the effects of it. It would be inconvenient, but according to the US government, detonating 500 tons of TNT would be similar (In blast effects only) to detonating a 1 kiloton nuke, and they specifically say it’s all to do with the speed at which that energy is released.

Which leads to my proposal: for blasts that are comparable (or greater) in speed to nukes this formula would apply (the formula proved valid for a fireball with a speed of ~13 km/s). That being because the formula is entirely dependent on the size and speed of the explosion, not the contents of said explosion. The explosion should also be large with a massive shockwave, like the magnitude of high speed explosives this formula was derived from.
 
IIRC we had a debate about this or a similar formula before. Long story short it's a formula won by dimensional analysis, which makes it a good estimation but generally using the usual (TNT & nuke based) explosion formula we have, which are based on experimental considerations, should be more certain.
Given, the formula would be useful if we could figure out the appropriate constants for magical explosions (dimensional analysis tends to not consider dimensionless constants), however I see no practical way of doing that. So we can just assume the constants for regular explosions for which we already have ways to quantify them.
So keeping it at the formula we have seems like a better idea.
 
Y'all had a debate about Taylor's formula, Cook fixed it aka made it accurate.
I see. Cook seems to have found an analytic expression for a priorly numerically found constant, but otherwise, it is still the same formula in my understanding.
The only constant, gamma, is a specific heat ratio of air, has nothing to do with magical explosions.
No no, I'm not saying that the constant I'm talking about is currently written in the paper. Dimensional analysis works by figuring out the formula of something by considering which things could influence the result and then arranging them in a fashion that the units/dimensions (I mean stuff like kg, m, s etc.) of the things in the formula match the end result. That why things without such units, i.e. dimensionless constants, don't appear in such considerations.

I get that the results of this formula are lab tested to fit regular explosive (well, I actually haven't read the entire paper so I don't know how well they are tested), but what I am saying is that they don't inherently are more accurate for magic.

In other words, this formula is at best as good as other experimentally found explosion formula. (or at least I have no reason to believe they are better)
Of course, there are many legitimate explosion formulas. Our current and this one are by far not the only ones. However, I would rather have one standard explosion formula than using many and getting multiple contradicting results.
 
In other words, this formula is at best as good as other experimentally found explosion formula. (or at least I have no reason to believe they are better)
Of course, there are many legitimate explosion formulas. Our current and this one are by far not the only ones. However, I would rather have one standard explosion formula than using many and getting multiple contradicting results.
Cook's formula has a proven accuracy for high speed explosions of within ~2% error. If it has tested accuracy I see no reason not to use it. Especially, if by your own admission the formula is perfectly fine, which I believe it is.

Our current formulae are good when there's no associated time frame, but Cook's is built for blasts that have a time frame to use.
 
What makes our both explosions formulas better than any other formula? It's more "safe"?
Didn't say they were better than every other formula. I just said I see no reason other formulas are better than them.
Cook's formula has a proven accuracy for high speed explosions of within ~2% error. If it has tested accuracy I see no reason not to use it. Especially, if by your own admission the formula is perfectly fine, which I believe it is.
As said, when we have multiple legitimate formula, I would much prefer to use a specific one to evaluate explosions, instead of having 5 different formula where it gets picked at random which one is used.

Btw. you said this is for ground explosions, but I think the paper actually suggests air burst?
Equations for the expansion rate of air burst fireball and its energy as function of its time and radius

Well, I suppose I might be fine with using this formula in addition to our usual ones in cases where the usual ones can't be accurately used for some reason.
However, I would tie two restrictions to its usage.
  1. The time factor needs to actually be known. Meaning no guessed timeframe and no cinematic time either. It needs to be actually stated or in some way measured.
  2. Seeing how attack speed is a separate speed rating (and fiction likes to keep its speed rating separate from power) the result must display an amount of destruction reasonable for the suggested energy. If an explosion is ultra-fast, suggesting megatons of yield, yet fails to damage the window in the building right next to it, this shouldn't be used. In practice that can be controlled by using our usual explosion formula to figure out in which range things should be heavily damaged. Given, this doesn't need to match precisely, but it shouldn't be off by a huge factor either.
 
  1. The time factor needs to actually be known. Meaning no guessed timeframe and no cinematic time either. It needs to be actually stated or in some way measured.
  2. Seeing how attack speed is a separate speed rating (and fiction likes to keep its speed rating separate from power) the result must display an amount of destruction reasonable for the suggested energy. If an explosion is ultra-fast, suggesting megatons of yield, yet fails to damage the window in the building right next to it, this shouldn't be used. In practice that can be controlled by using our usual explosion formula to figure out in which range things should be heavily damaged. Given, this doesn't need to match precisely, but it shouldn't be off by a huge factor either.
This seems agreeable and makes sense.
 
Btw. you said this is for ground explosions, but I think the paper actually suggests air burst?
The air burst stuff is post the formula and related but it uses the principles from Taylor, whereas Cook's formula itself isn't used specifically for air burst iirc. The Cook formula that's relative is primarily just for the fast fireball, the air burst stuff is a separate derivation.
 
If a character A says that a character B with FTL speed cannot escape an explosion with a radius of 5 kilometers, can it be used for 1 second? Basically a Low-End
This shouldn't be used for FTL shockwaves at all, so no.
Even for not specifically FTL things, that would be calc stacking or more specifically hiding calculations.
The air burst stuff is post the formula and related but it uses the principles from Taylor, whereas Cook's formula itself isn't used specifically for air burst iirc. The Cook formula that's relative is primarily just for the fast fireball, the air burst stuff is a separate derivation.
Are you sure? I admittedly have only read the Introduction and Method section yet, but it sounds like this assumes a spherical expansion of the shockwave i.e. airburst.
 
Are you sure? I admittedly have only read the Introduction and Method section yet, but it sounds like this assumes a spherical expansion of the shockwave i.e. airburst
I’m pretty sure, I’d have to reread the pdf, I wanna saw Cook’s formula just cares about the speed at which the actual explosion expands (ie radius and time)
 
A question: This new formula should only be applicable to explosions that are massive in scale and can do damage on par with IRL nukes while having the same speed as nukes (At least 13 km/s as said in the OP) at the bare minimum, correct?
 
A question: This new formula should only be applicable to explosions that are massive in scale and can do damage on par with IRL nukes while having the same speed as nukes at the bare minimum, correct?
Yes, explosions that are less than "atom bomb" level prolly shouldnt use this
 
Don’t mean to bash any chance of getting this accepted, but the “gamma” part gets a bit wonky the higher you go. While 1.4 was perfect for getting the trinity bomb result, 1.67 was used in this calc to get the tsar’s yield, and both were basically perfect.
 
Don’t mean to bash any chance of getting this accepted, but the “gamma” part gets a bit wonky the higher you go. While 1.4 was perfect for getting the trinity bomb result, 1.67 was used in this calc to get the tsar’s yield, and both were basically perfect.
I wanna say that’s due to it being much colder in Russia on average.
 
I wanna say that’s due to it being much colder in Russia on average.
I mean I’ll take your word for it. Though I have to ask, could this be used for smaller explosions that occur under water? No other formula has the density variable, so it could be useful for some SB calcs.
 
That’s have to be toyed around with. I’d have to look into the specifics of deriving the gamma quantity because I feel like water would change that. Part of me wants to say yes since the formula is dependent on size and time but then the other part of me is skeptical cuz it was derived from out of water explosions.
 
Didn't say they were better than every other formula. I just said I see no reason other formulas are better than them
Let me rephrase. What makes you think that our current formulas are usable and others are not? Why do we use the current ones and not something else?
 
Anyway, anyone figure out how to use Cook's Formula underwater?
 
I just wanted to out that the math should be done using the first frames of the explosion. The reason being is that the radius affects the result far more than the time does. This means that the same explosion could have wildly different results.
Ex: An explosion occurs at sea level, where the density of air is 1.225 and gamma is 1.4. Said explosion moves at 100m/s.

8pi x 1.225 x 50^5/(75(0.4) x 0.5^2) = 1282817000 joules
8pi x 1.225 x 100^5/(75(0.4) x 1^2) = 10262536001 joules
Same explosion, just different points.
 
So I’ve been reading the article where the formula comes from, and a particular detail caught my attention:

These results are useful to interpret the filmed test fireball expansion rates in terms of energy, to analyze supernovae explosions, and in constructing the equation for the arrival time of the blast wave at any distance.

Supernovae are practically outer space explosions. So I’ve been thinking: can this formula be used for outer space explosions that have (but realistically shouldn’t) shockwaves?
 
So I’ve been reading the article where the formula comes from, and a particular detail caught my attention:

These results are useful to interpret the filmed test fireball expansion rates in terms of energy, to analyze supernovae explosions, and in constructing the equation for the arrival time of the blast wave at any distance.

Supernovae are practically outer space explosions. So I’ve been thinking: can this formula be used for outer space explosions that have (but realistically shouldn’t) shockwaves?
I believe that a Godzilla calc did that and it was accepted.
 
I just wanted to out that the math should be done using the first frames of the explosion. The reason being is that the radius affects the result far more than the time does. This means that the same explosion could have wildly different results.
Ex: An explosion occurs at sea level, where the density of air is 1.225 and gamma is 1.4. Said explosion moves at 100m/s.

8pi x 1.225 x 50^5/(75(0.4) x 0.5^2) = 1282817000 joules
8pi x 1.225 x 100^5/(75(0.4) x 1^2) = 10262536001 joules
Same explosion, just different points.
So what do all those factors stand for?
 
I know this was already accepted

But I gotta ask, on the point of having a known timeframe and absolutely no guessing

If a statement like, would’ve instantly annihilated everything in (x radius) is used would a time frame of 5 or 2 seconds be ok

Since a word like instant should make a short timeframe reasonable
 
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