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Basic Exterior Ballistics Theory Ballistics theory states that bullets of different diameters and weights, but of similar shapes, decelerate in constant proportion to each other. That is, if a bullet's deceleration is twice that of another similarly shaped bullet at one velocity, then that bullet's deceleration remains twice that of the other bullet at any other velocity. Thus, by knowing a bullet's deceleration at one velocity, its deceleration at a second velocity can be predicted by knowing the deceleration of a similarly shaped bullet at both velocities. To make the theory useful, a number of "standard" bullets have been test fired thousands of times to determine their deceleration at many velocities. This experimental data is then mathematically analyzed and a curve (function) that best matches the experimental data is produced. Deceleration is the result of drag due to air resistance, so these functions are called drag coefficient functions. When these functions are expressed in the form of tables, we call them drag coefficient tables or just drag tables for short. The two main forces acting on a bullet in flight are gravity and drag due to air resistance. In horizontal flight, gravity causes the bullet to accelerate (drop) toward the ground while drag causes the bullet to decelerate (slow down). While unconventional, it's useful to express drag in gravitational units (G). A force of one G is equivalent to the force exerted by gravity on an object near the surface of the earth. Ignoring air resistance, a free falling object accelerates at 32.17 f/s². That is, after one second of falling, the object's velocity is 32.17 f/s, after two seconds its velocity is 64.34 f/s (2 x 32.17), after three seconds its velocity is 96.51 f/s (3 x 32.17), and so on. Likewise, an object thrown straight up decelerates at 32.17 f/s². Incidently, the names of most drag functions start with the letter G in honor of the Gâvre Commission in France (1873-1898) and not because of gravitational units. The figure below shows the standard drag functions G1, G5, G6, and GL plotted as Deceleration in G's (force of gravity) for velocities from Mach 0.5 to Mach 4.5. Mach 1 is the speed of sound, which is 1120.27 f/s at standard Metro conditions (59° F and 78% RH).
Each of the drag functions is based on a standard projectile of a particular size and shape. For example, the G1 function is based on the Krupp projectile which has a flat base, and is 3.25 calibers long with a 2 caliber ogive tip. The speed of sound in air is affected by temperature and somewhat by humidity, and as temperature and/or humidity increase the speed of sound increases. While pressure (altitude) by itself doesn't affect the speed of sound, for a given temperature, the ratio of water vapor in the air increases as pressure decreases, and thus, the speed of sound increases slightly with altitude. The air resistance felt by a bullet depends both on the density of the air and the speed of sound through that air. This is why drag functions are defined in terms of Mach number rather than absolute velocity. At standard conditions, a test bullet's deceleration and corresponding Mach number can be calculated from an initial velocity (V1) and another velocity (V2) at a known distance down range using the following formulas:
The values of V1 and V2 can be obtained experimentally using two chronographs. The Ballistic Coefficient (BC) of the test bullet is the ratio of the standard bullet's deceleration divided by the test bullet's deceleration at the same Mach number.
If the test and standard bullet's shapes are similar, then the BC will remain nearly constant over a wide range of velocities. Thus, the suitability of a particular drag function can be gauged by calculating a second BC for the test bullet using significantly different values of V1 and V2. To the degree that the part of the drag function spanned by V1 and V2 is nonlinear, an error is introduced into the calculated Mach number, and thus, into the calculated BC. This error as well as those introduced by non-standard conditions and winds can be corrected for by methods employed in the program, but which are beyond the scope of this help. It's also a lot quicker to let the program calculate BC. see Ballistics Model |
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