playing with dimensions

Many occupations develop tricks to get "good enough" answers question.  These range from rules of thumb to shortcuts that are based on some insight.  In physics dimensional analysis - checking to see the technique or answer has the right dimensions - is important. After all - you don't want to measure someone's height and come out with an answer in pounds.   

 

The common shorthand is to use square brackets to denote the dimension of a physical quantity.   For a distance r [r] = L, velocity [v]  = L/T, g (the acceleration of gravity) [g] = L/T².  Generally M is the dimension for mass, L for length, and T for time. 

 

With a bit of additional information you can make quick estimates and even gain some insight.  Consider two athletes of similar build and physicality, but different heights.  One is six feet tall, the other five feet.  If you want to compare the power to weight ratio all you need to know is the power a muscle generates is proportional to its cross sectional area.  Their weights scale as the cube of their height, so  [power/weight] = L²/L³ or 1/L.  The shorter athlete has a twenty percent power to weight advantage.   You can use this type of analysis and freshman level physics to sort out why different sports dictate very specific body types at the elite levels of sport.  

 

Techniques like these are used during the early and very playful part of asking questions.  They can tell you if your approach is plausible or if it's going to be wrong.  Plausible doesn't mean right, but it's very useful to recognize what can't work so you can try something else.  It's a lovely way to learn by failing gently.  With time and experience you develop an intuition for what might work. 

 

One of these quick calculations is legendary.  A series of photos of the Trinity after the end of WWII.   Four stages of the explosion with a length and time indicators.  Details like yield were classified.  A British physicist didn't realize calculations were classified and did his own back of the envelope calculation in a few minutes.  His estimate was published and caused a kerfuffle worrying that secret information had been leaked.  

 

There wasn't a leak. The truth is many of the physicists involved made quick estimates that were good enough - enough to tell them the device's yield was about what they thought it would be.  They just didn't publish for obvious reasons. 

 

Here's my ten minute estimate using dimensional analysis and a bit of physical intuition. I'm avoiding a very difficult set of differential equations with some very messy boundary conditions - not to mention some quantities I have no way of knowing as an outsider.  To make it as simple as possible consider a spherical blast. The ground will have a minimal on the radius of the blast in the atmosphere, so ignore it.  The blast creates an expanding bubble that has to push against air outside the bubble. The most important terms in any expression will include the radius of the blast, the density of the air outside the blast, and the energy of the device.  There are many other things going on - this is just an attempt to see if a very simple high level relation of the relevant variables can be found in a few minutes.

 

Basically rearrange these terms dimensionally to see if you can find radius as a function of time after the device was set off.   Remarkably a simple relationship emerges.  So why not plug some numbers in?   The time and radius can be found in the photos and the density of air is well-known.    My radius measurement is crude - I know the device was on a tower and found a point that looked like the center.. draw some lines on a piece of paper, put it up to the picture and estimate 90 meters... perhaps a bit less.  A bit of arithmetic and a yield of equivalent to fifty thousand tons of TNT falls out.   

 

(pardon the handwriting, but it's much quicker and more natural than trying to typeset it)

 

 

The estimate isn't that far off - it gives you a good sense how the radius scales as the one fifth power of the blast's energy and to the two fifths power of time.  Those relationships are more illuminating than the estimated yield.  Since there are four images you could do something a bit more elaborate measuring all of the photos and making a plot of the growth of the fireball.