This page attempts to explain the uses of the various quantities that I have used in ballistics calculations, such as KD, G functions, CD, etc. The equations here deal mainly with the acceleration of a bullet when treated as a point mass. References and a list of variables are at the bottom of this page.
The force on the bullet due to drag, using K_{D}, can be given as:
where K_{D} is a function of mach number. In terms of K_{D}, the G function can be written as
so that the drag force on the bullet may be written as
Defining the ballistic coefficient, C, as
[M is the mass, and not the weight -- to convert to the standard sectional density, multiply by g, the acceleration due to gravity] we can rewrite the previous equation in terms of the bullet acceleration, G function and C
To find a trajectory, we have to sum all forces (and therefore all the accelerations) on the bullet and integrate the resulting equation. For point mass trajectories, the only other force we need to account for is due to gravity:
This expression can be integrated from the muzzle to the target to obtain a three dimensional point mass trajectory.
In terms of the newer ballistic nomenclature, C_{D}, we may rewrite K_{D} as
so that the the equation for G is
The acceleration of the bullet, due to drag, in terms of C_{D} is
NOTE: that in the preceding equation, the C_{D} is that of the "standard" bullet, for which K_{D} was defined.
For the C_{D} of the given bullet (as output in the Drag calculations, and McDrag program)
or plugging in for the cross sectional area (or reference area)
and finally with the sectional density
a | total vector acceleration (du/dt) | a_{D} | vector acceleration due to drag |
C | ballistic coefficient | C_{D} | drag coefficient |
d | bullet diameter | F | vector drag force |
g | vector gravitational acceleration | G | "G" function |
i | form factor | K_{D} | drag coefficient |
m | mach number | M | bullet mass |
p | atmospheric density | p_{0} | sea level atmospheric density |
π | 3.14159... | SD_{M} | sectional density (with mass) |
s | cross sectional area (π d^{2}/4) | v | vector velocity, | v | = (v·v)^{1/2} and v = u - w |
w | vector wind velocity | u | velocity of bullet relative to ground |
The Effect of Wind on Flat-Fire Trajectories, Robert L. McCoy, BRL Report Number 1900, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, August 1976, [ADB012872]
Exterior Ballistics, McShane, Kelley and Reno, Univeristy of Denver Press, 1953.
Exterior Ballistics of Small Arms Projectiles, E. D. Lowry, Research Dept., Winchester-Western Division, Olin Mathieson Chemical Corporation, May 4, 1965.
Sierra Rifle Reloading Manual, 4th Edition, Sierra Bullets, L.P., 1995.