By Wm. C. Davis, Jr.

   Sir lsaac Newton in 1686 published a treatise called "The Mathematical Principles of Natural Science," and in it he set forth three fundamental laws of nature which govern the relationship between force and motion. The science of ballistics, and indeed the whole science of mechanical physics, are based on these three laws. Since this is not a lesson in mechanics, we will not dwell on Newton's three laws of motion here. It can be shown, however, that Newton's second and third laws of motion lead to the principle of conservation of momentum. From this principle, it follows that the momentum imparted to the gun upon firing must always be equal in magnitude, and opposite in direction, to the momentum imparted to the bullet and the powder gases that are expelled from the bore. Since the momentum of a body is the product of multiplying its mass by its velocity, the fundamental equation of recoil can be expressed as follows:

   MG * VG = MB * VB + MC * VC

      where MG = mass of the gun
                VG = velocity of the gun
                MB = mass of the bullet
                VB = velocity of the bullet
                MC = mass of the powder charge
                VC = velocity of the powder charge

   The masses of the gun, the bullet and the powder charge are easily determined by weighing them. The velocity of the bullet is easily measured by a chronograph, and the velocity of the gun can be determined by various means, the most convenient of which is the recoil pendulum. The only quantity not directly measurable in the fundamental recoil equation is therefore the effective velocity of the powder charge, VC. Since the other quantities can all be measured, their values can be substituted in the equation, and the value of VC can thus be found for any particular experiment. This has been done a great many times, for a wide variety of guns. For a variety of cannon, it has been found that the value of VC is about 4700 f.p.s., and that number is sometimes quoted in formulas for calculating the recoil of small arms as well, but experience shows that it is too high for small arms.
   Experiments carried out by Hitchcock and Kent at Aberdeen Proving Ground in 1929 for various .30 cal. guns and loads showed values from 3710 to 4115 f.p.s. for VC. The British Textbook of Small Arms (1929) reports measurements made on the .303 British rifle with Mark VII ammunition which gave a value of 3725 f.p.s. for VC. I made recoil measurements on some standard .30 cal. rifles and some experimental .22 cal. high-velocity rifles at Aberdeen Proving Ground in 1953, and found values of VC from 3920 f.p.s. to 4100 f.p.s. I have since had occasion to make such measurements on a 7 mm Rem. Mag. rifle with various loads, and found values of VC from about 3700 f.p.s. to 4300 f.p.s. Col. E.H. Harrison wrote in the NRA Firearms and Ammunition Fact Book that the evidence he had studied suggested a typical value of about 4000 f.p.s, for VC in small arms, and indeed that agrees very well with the data that I have seen for guns firing smokeless-powder charges. There is not much reliable information available on experiments made with blackpowder loads, but such information as there is suggests a velocity of about 2000 f.p.s. for VC. In the following equations, 2000 should therefore be used instead of 4000 for the value of VC when calculating the recoil of blackpowder loads. This does not imply that blackpowder loads produce less recoil than do smokeless loads, as can be seen from working out examples. Blackpowder loads produce greater recoil than do smokeless loads that give the same muzzle velocity to the same projectile, because the use of blackpowder requires that a much heavier charge be used.
   It should be noted that formulas sometimes quoted for calculating recoil have set the velocity of the powder gases equal to some constant, such as 1.5, multiplied by the muzzle velocity of the bullet.
   However, that agrees well with experiment only for muzzle velocities of about 2600 to 2700 f.p.s., in which case 1.5 times the muzzle velocity is about 4000 f.p.s. That approach is evidently flawed in principle as well, since it implies that shortening the barrel, which would reduce the muzzle velocity of the bullet, would also reduce the escape velocity of the powder gases, whereas that manifestly is not the case. Shortening the barrel increases the pressure of the gases at the muzzle, which would serve to increase rather than to reduce the velocity at which they escape from the bore. In the fundamental recoil equation, the term MG * VC is the momentum imparted to the gun, while MB * VB and MC * VC are the momenta of the bullet and the powder charge respectively. The momentum of a body is numerically equal to the impulse which produced the momentum, and the term MG * VG is customarily called the recoil impulse, represented by the letter "I", so the equation becomes:

   I = MB*VB + MC*VC, where I is the recoil impulse.

   If 4000 f.p.s. is taken as the effective velocity of the escaping powder gases, and constants are introduced to adjust for the units of measurements customarily used, the fundamental recoil equation for smokeless-powder loads becomes:

   I = (WB*VB + 4000*WC)/225400

   where I = recoil impulse in lb.-sec.
         WB = weight of the bullet in grains
          VB = velocity of the bullet in f.p.s.
         WC = weight of the charge in grains

   The recoil impulse depends essentially upon the performance of the ammunition, and it is independent of the weight of the gun. It is the basis on which the recoil characteristics of different cartridges and loads are sometimes compared, uncomplicated by consideration of the differing weights of guns. It is also the fundamental equation from which other characteristics of recoil are derived. There are two other characteristics of recoil which are perhaps more important in determining the "kick" or sensation of recoil. One is the free-recoil velocity, which can be found from the following equation:

   VG = 32.2*1/WG

      where VG = velocity of the gun in f.p.s.
               WG = weight of the gun in lbs.

   The free-recoil velocity of the gun can make it unpleasant to shoot, even if the recoil impulse is not excessive. The British Textbook of Small Arms (1929) says that, for guns weighing 6 to 7 lbs. a free-recoil velocity of 15 f.p.s. "…has long been established as a maximum above which gun-headache is sure to ensue." That is also the reason that the recoil of a light gun may be perceived as a sharp and uncomfortable jab to the shoulder, whereas a heavier gun producing equal free-recoil energy is perceived as producing a slower push that is not so uncomfortable. The third characteristic of recoil, and perhaps the most meaningful single predicter of the "kick" sensation, is the free-recoil energy. It is found from the following equation:

   EG = WG*VG²/64.4

   where EG is the free-recoil energy in ft.-lbs.

   It should be mentioned that the "kick" or sensation of recoil is somewhat affected by factors not taken into account in these calculations. In particular, autoloading guns often produce a much softer "kick" sensation than do fixed-breech guns. This is because the spring-loaded recoiling parts are being driven rearward relative to the remainder of the gun during recoil, and their rearward momentum is not transferred to the buttstock or grip of the gun very suddenly, but is "fed back" more slowly during the recoil and counter-recoil movement of the spring-loaded parts. This is a real physical effect, and it can be confirmed by suitable instrumentation. It is sometimes supposed that the use of a "slower" powder can also soften the "kick" sensation, but that is debatable, because the burning of the powder is completed, in any case, before the gun has moved more than about 1/10", and thereafter, the speed of powder burning could make no difference in the movement of the gun. There are also some physical factors such as the drop of the stock, the slope of the comb, the size and resiliency of the buttplate, the security of the shooter's grip on the fore-end, or the size and shape of the grips on a handgun, which affect the sensation of recoil. Finally, of course, there are some purely psychological factors which may strongly affect the subjective perception of "kick", apart from any real physical phenomena. A shooter who has been thoroughly convinced that his gun will not "kick" hard may perceive after firing it that he has hardly been "kicked" at all, whereas a shooter expecting to be "kicked" very hard may fire the same gun and perceive that his worst expectations were fulfilled. This gives rise to countless differences of opinion, and some vociferous arguments, about the relative recoil effects of various guns, cartridges, and recoil-reducing devices.
   The accompanying tables list recoil characteristics for some typical rifle, handgun and shotgun loads. For readers who remember their algebra, the equations above can readily be solved for other guns and loads in which they are interested. A simple pocket calculator will do all of the necessary arithmetic. For those who have access to a personal computer, the task is made easier by the accompanying computer program. It is written in the BASIC dialect used by "IBM-compatible" computers. Readers are hereby granted permission to copy the program for their own personal use, but it is copyright material, and copying it for commercial exploitation is forbidden by copyright law. If you are not familiar with copying and running programs in BASIC, you should read the article titled "How to Compute in BASIC''', found elsewhere in NRA Firearms and Ammunition Fact Book.