Friday, February 26, 2010

Physics Problem of the Week

So... I'm reading the current Dick Tracy comic strip (online, of course), and lo and behold, a physics problem comes up. The current story line, which has been running since Thanksgiving, involves a Stradivarius violin that's been loaded with plastic explosive, rigged to go off when the violinist hits a high C (most likely a C-4), and sent to a fancy reception at a foreign embassy. We don't yet know why somebody's going to such great lengths, or such a roundabout plot, to assassinate the ambassador (perhaps that will be revealed later), but we do know that as of last Saturday, Tracy had grabbed the Strad and chucked it into the swimming pool seconds before it went off with (and I quote) an enormous "FLOOOOM!"

In the wake of the blast, several questions came up:
  • Why would someone pay two million bucks for a Stradivarius, when a cheap high-school orchestra violin would have exploded just as well?
  • Why is there and embassy in Naperville?
  • What are those two big hand-like things sticking out of the pool during the "FLOOOOM!"? (Could the explosion have aroused the Ancient Slime Monster foretold by Mayan prophecy?)
  • Why is there a swimming pool in the embassy's formal dining room?
Of course, these are questions that can't be answered by physics (or any science, for that matter). But two other questions came up, questions that are amenable to science, and they center around what happened to the water:
  • Since the water seems to have simply disappeared in later strips, did the explosion vaporize it?
  • Or, if the explosive lacked the ooomph (that's the technical term) to vaporize the water, could it at least actually blow the water out of the pool.
These questions we can answer with a little chemistry and physics.

A quick trip to the internet divulged two important bits of information: C4 plastic explosive is mostly something called RDX, with a chemical formula of C3H6N6O6. It's a pretty interesting molecule, with a structure kind of like a snowflake--an inner hexagonal ring of alternating carbon and nitrogen atoms, and branches off that ring. Each carbon binds to two hydrogen atoms (that's pretty standard, like gasoline), and each of the three nitrogens is hooked up to an NO2 group, which is sort of a molecular menage a trois, in which the two oxygen atoms are hooked up with the nitrogen for now, but would happily take a better offer if one came along. The RDX molecule is energetic and only kind of stable, and when it breaks down into N2, H2O, CO2 and some loose carbon soot, it gives off a lot of energy. How much? Well, according to a paper I found on the internet, where the author analyzed those complicated "resonance" bonds in the NO2 groups, RDX liberates 10 million joules of energy per mole when it goes off.

According to another website on military stuff, the "standard" brick of C4 explosive is half a kilogram, and is about 90 percent RDX (the rest is binders and stuff). So, given the molar mass of RDX is 222g, that brick is just about exactly two moles... so when the Stradivarius goes off, 20 million joules get released.

Is that enough to vaporize the water in the pool? Not even close. It takes a lot of energy to boil water--something over 2 million joules per kilogram, so the 20 million joules released by the explosive in the Strad would only boil away about ten liters. Barely enough energy to boil the champagne.

But is it enough to empty the pool? Well, let's look at the pool and make some estimates. Looks to be about ten meters in diameter, and let's assume it's about two meters deep. That gives a volume of 25 times pi times 2, or 157 cubic meters. A cubic meter is 1000 liters, or 1000 kilograms, so the mass of water in the pool is 157,000 kilograms. So, when the Strad goes "FLOOOOM!", we've got 20 million joules acting on 157,000 kilograms. From these numbers, and the formulas for kinetic and gravitational energy, we can figure out how fast the water will be flung out of the pool, and how high it will splash.

First, how fast: the formula is Ek=0.5mV**2 (sorry, but Blogger won't typeset exponents). Rearranging the terms to solve for velocity, we get V=sqrt(2Ek/m), and if we plug in the numbers that's V=sqrt(2*20,000,000/157,000)=sqrt(255)=16 meters per second, which is about 35 miles an hour.

Second, how high: the formula is Eg=mgh, where g=the acceleration of gravity, 9.8 m/sec**2. Rearranging the terms to solve for height, we get h=Eg/(mg), and plugging in the numbers gives us h=20,000,000/(9.8*157,000)=20,000,000/1,530,000=13.1 meters, or about 43 feet. That's a pretty good lift, but... it's been three days since the Strad went off, and the water still hasn't come back down. Time must work differently in the Dick Tracy universe... but that's relativity, and that's a problem for another day.

1 comment:

David said...

Lovely analysis. Question: could the Strad have even played a C with any resonance had it been filled with explosives? And wouldn't the violinist noticed the weight gain in their instrument? I mean, violins are lightweight, violence, in the firm of an explosives-packed violin, would be quite heavy.