Various mathematical equations were developed to relate muzzle velocity to barrel length, but one of the simplest relationships was developed by Homer S. Powley. He defined the relationship between muzzle velocity and barrel length, as one giving muzzle velocity (v) as a function of charge weight C, bullet weight B and expansion ratio (R). Here the expansion ratio is defined as the ratio of the barrel volume plus cartridge volume (total volume of the gun) to the cartridge volume. The equation relating to these factors is represented by:
v = K[C(1- R^-.25)/ (B + C/3)]^.5
Where, v is in f.p.s., C and B are in grains and K is a constant that depends on chamber pressure and other factors in the gun. The expansion ratio is dimensionless. From this equation, it can be seen that for a given gun, with a given powder charge and bullet weight, the muzzle velocity is dependent only on the expansion ratio. By cutting off the barrel the barrel volume is reduced, thereby reducing the expansion ratio. The relationship can be represented by:
F = [(1-R2^-.25)/(1-R1^-.25)]^.5
Where, F is the correction factor to correct the muzzle velocity at expansion ratio R1 to that of the reduced expansion ratio R2 (shorter barrel). For example, a rifle chambered for the .223 Rem. cartridge, has a 24" barrel, and fires a 50 gr. bullet at 3,080 f.p.s. when loaded with 25.1 gr. of IMR 3031 powder. The expansion ratio is 8.5. What is the muzzle velocity if the barrel is shortened to 22"? The new expansion ratio is 7.8. Plugging these values in the equation we get:
F = [(1-7.8^-.25)/(1-8.5^-.25)]^.5
= 0.9846
v = 0.9846 X 3,080
= 3,032 f.p.s.
The new velocity for the 22" barrel is 3,032 f.p.s. The velocity loss for removal of two inches of barrel is 48 f.p.s. (3,080 - 3,032).
And there you have it! There will be a test...