Skulls in the Stars

During my first evening in San Antonio, I sequestered myself in my hotel room to polish up my presentation.  Fortunately, there was a Mythbusters marathon on the Discovery Channel at that time, so I was able to keep myself marginally sane by watching the ‘Busters abuse places, things, and themselves for the cause of science.

One of the episodes that played during the marathon contained the “finger in the barrel” myth — the idea that a person can stick a finger in the barrel of a rifle or shotgun as it fires, causing the barrel to split like a banana peel without harm to the finger!  The initial investigation of the ‘busters clearly demonstrated that a finger would certainly be lost in the attempt, and that a barrel would not split in the manner suggested.  An updated investigation two years later, however, demonstrated that a rifle barrel could be split if sufficiently weakened by use.

In a remarkable case of serendipity, the next evening I was browsing the Proceedings of the Royal Society of Edinburgh and came across an article with the title, “On the bursting of firearms when the muzzle is closed by snow, earth, grease, &c.”!  The article, by Professor George Forbes, is a theoretical explanation of the bursting of firearms and was published in the 1878-1879 session of the Royal Society, meaning that Forbes’ investigation was some 130 years before the Mythbusters!  The calculation and explanation are short and entertaining, and I thought it would be fun to take a look at them.

The article begins as follows:

It is well known that if an ordinary fowling-piece, charged with shot or ball, have touched the ground or snow, so as to close the muzzle of the gun, or if the muzzle of the gun be in any way artificially closed with grease or other substances, the fowling-piece is certain to burst at the muzzle when it is discharged.  This would not be the case if, instead of firing a shot, a piston were driven up the tube by hand.  In this case the compressed air would drive out the opposing plug, which offers but a very feeble resistance to the internal pressure.  These facts, thoroughly well authenticated, have not, to my knowledge, received a satisfactory explanation, though a clear idea of the conditions of the case is all that is required to explain this, at first sight, anomalous behaviour.

Of course, this article does not mention a “banana peeling” of the barrel, the description of a barrel being “certain to burst at the muzzle” seems pretty close.  It’s rather intriguing that Forbes refers to such an occurrence as “well known”; though the Mythbusters had to work very hard to disfigure a barrel, it was apparently quite easy to do with the weapons of Forbes’ time.

The puzzler in this problem is that air forced into the barrel by piston will push out a plug just fine, while an actual gunshot will destroy the barrel before pushing out an identical plug.

The explanation lies in the fact that the charge travels along the bore of the gun, if not with the same velocity as, at least with a velocity comparable to, that of the transmission of pressure through the air, i.e. the velocity of sound.  Thus, as the charge advances along the barrel it is continually compressing the air immediately in front of it; but this pressure gets no relaxation by expansion into the front part of the barrel.  The compression, of course, generates heat in the air, which increases the velocity of sound through it.  But this does not affect the question in its general bearings.  It is sufficient to notice that the snow, &c., is driven out with the full velocity of the charge (neglecting the weight of the snow-plug compared with that of the charge).  But before the plug can be driven out with this great velocity the pressure behind it must be very great.

The hypothesis, it seems, is that the air pressure grows so quickly that it reaches the point of shattering the barrel before pushing out the plug.  Of course, such a hypothesis needs to be backed up by some evidence, and Forbes provides a very simple calculation to justify his claims; we summarize it here (and fix a mistake or two in the math along the way).

It is assumed that the “snow-plug” of mass $latex m$ leaves the gun at velocity $latex v$ comparable to the speed of the bullet.  On leaving the gun, it has kinetic energy $latex w$ equal to

$latex displaystyle w = frac{1}{2}mv^2$.

This work is produced by the pressure $latex p$ of the gas in the barrel; denoting the cross-section of the bore as $latex A$ and the length of the snow-plug at $latex b$, and noting that work = force times distance, we have

$latex displaystyle p A b = frac{1}{2}mv^2$.

Solving for $latex p$, we have

$latex displaystyle p = frac{mv^2}{2Ab}= frac{1}{2} rho v^2$,

where $latex rho = m/Ab$ is the density of the snow-plug. (This number is a factor of 2 different than Forbes’, as he assumed that the speed was created over a distance $latex b/2$.)

Forbes assumed a velocity $latex v$ of 1000 feet per second; in terms of meters/second, we have

$latex v approx 300mbox{ m/s}$.

The density of water at freezing temperature is roughly

$latex rho approx 1000 mbox{ kg/m}^3$.

Putting these together, we have

$latex p = 45 mbox{ mega-N/m}^2$!

A metric ton is 1000 kg; if we convert our number into metric tons per square cm, we have

$latex p = 0.45 mbox{ tons/cm}^2$!

This is a pretty huge pressure — a half ton per square centimeter.  Forbes’ calculation, in English units, results in 7 (English) tons/square inch.  My calculation, neglecting his extra factor of 2, would be about 3 (English) tons/square inch.

Forbes summarizes his conclusions with the following:

A pressure which the muzzle of a shot gun is not constructed to withstand, and the theory shows that this great pressure can be produced even by a plug of snow or grease of the shortest length movable inside the barrel with the greatest facility.  If the velocity of the ball or wad be less than that of sound the snow-plug is not driven out quite suddenly, and if the velocity be small enough the snow-plug is driven out before the ball or wad reaches the muzzle.

Is his calculation correct?  It’s hard to say, as it is a great simplification of a problem with many factors.  There does not seem to be anything fundamentally wrong with the basic physics equations he uses, however.

At the very least, Forbes’ paper shows that science-types have long been intrigued by exploding things!

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G. Forbes, “On the bursting of firearms when the muzzle is closed by snow, earth, grease, &c.” Proceedings of the Royal Society of Edinburgh 10 (1878-1880), 254-256.