The internal ballistics of a firearm or artillery piece considers the pellet, bullet, or shell motion while it is still inside the barrel. In general, deriving the muzzle speed of a gunpowder firearm from first principles is difficult because powder combustion is fast and it very rapidly raises the temperature of gas (generated by gunpowder deflagration, or burning), which greatly complicates the analysis. A simple case is provided by air guns, for which we can make reasonable approximations that permit a derivation of muzzle speed. It is perhaps surprising that muzzle speed depends upon barrel length (artillerymen debated this dependence for centuries, until it was established experimentally and, later, theoretically1). Here we see that a simple physical analysis, accessible to high school or freshmen undergraduate physics students, not only derives realistic muzzle speed but also shows how it depends upon barrel length.

Topics
Projectiles, Ballistics, Deflagration, Combustion, Education
1.
See, for example, Bert S. Hall, Weapons and Warfare in Renaissance Europe (Johns Hopkins University Press, Baltimore, 1992);
Captain J. G. Benton, Ordnance and Gunnery; Compiled for the use of the Cadets of the United States Military Academy (D. van Nostrand, New York, 1862).
2.
Thus our calculation is based upon isothermal gas expansion. This contrasts with the analysis of Mungan, who assumed adiabatic expansion. See
Carl E.
Mungan
, “
Internal ballistics of a pneumatic potato cannon
,”
Eur. J. Phys.
30
,
453
–
457
(
2009
).
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3.
The difference between projectile diameter and barrel diameter for an old‐fashioned musket is called windage. Windage was unavoidable for these muzzleloaders, even though it wasted gas pressure, because a tight‐fitting ball could not be rammed down the barrel. An air rifle pellet expands to fill the windage gap, much as did a “Minié ball”—the bullet fired from a Civil War‐era rifled musket. See, for example, Barton C. Hacker, American Military Technology (Johns Hopkins University Press, Baltimore, 2006), p. 21.
4.
From the twist of the rifling and the muzzle speed of a pellet, it is straightforward to determine the projectile angular speed— typically this is two orders of magnitude below the pellet translational energy. Powerlets (cylinders containing 12 g of liquid CO2 that are a common source of gas for low‐caliber air guns) last for about 40 shots before the pressure drops unacceptably, and so they expend less than 5 grains of gas per shot—a third of the weight of a .22 pellet.
5.
Spring‐loaded air guns are analyzed in an online article (home2.fvcc.edu/∼dhicketh/Math222/spring07projects/StephenCompton/SpringAirModel.pdf) in which MATLAB is employed to solve the equations.
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© 2011 American Association of Physics Teachers.
2011
American Association of Physics Teachers
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