**Connection to Classical External Ballistics**

Rifleist uses the term "classical" to refer to the use of tabulated drag functions for calculating bullet trajectories. In the centuries before velocities could be measured there was no alternative. But with velocity measures available other methods discussed here are possible.

With reference to the recent book by McCoy Ref. 2, his eq. 5.8 is dV/dT = -CMcC V^2. CMcC is the raw drag coefficient or function. This equation is the starting point for almost all external ballistics including bullets, cannons, and rockets. For projectiles the coefficient -CMcC = r a Cd/(2m) where r is air density, a is area, m mass, and Cd the drag coefficient or drag function of velocity. For rifles, -CMcC = r pi Cd/ (8Cb) where Cb is the usual ballistic coefficient and Cd is one of the standardized drag functions. Since -CMcC is a drag, higher Cb means lower drag and better flying.

dV/dT = (dV/dS)(dS/dt) = V (dV/dS) = -CMcC V^2 (13A)

From 13A, a property used by Pejsa in Ref. 3 is

(dV/dS) = -CMcC V = d/ds ln (V) (13B)

But a critical part of this analysis is the following, noting that dV/dS = 2AS+B :

-CMcC = (dV/dT) / V^2 = (dV/dS) / V = (2AS+B) / (AS^2 + BS + Vz) (13C)

Eq 13C shows that the raw drag is completely expressed by the velocity profile. Thus, the X and Y drag functions are coupled through the velocity profile, but not through each other. And, it should be noted that any polynomial function that fits the velocity profile could work at this stage. (In a previous section it is noted that higher order polynomials can work well in some circumstances.)

From the fitted parameters A and B and Vz and using Eq. 13C and the air density, etc, one can find the exact ratio of the drag coefficient to ballistic coefficient at each point on the trajectory. Other derivations have produced the following function of T:

-CMcC = dV/dT /V2 = (2A/R)*sin(R T + 2 atan(B/R) ) (13D)

Finally, for calculating trajectory tables, the integral of -CMcC over S is ln( A S^2 + B S + Vz)

We observe that for the many trajectories described by Eq. 1, the coefficient CMcC varies substantially during the flight, which is part of the difficulty with classical external ballistics for describing supersonic flight.

**Estimates of A,B from a Ballistic Coefficient**

From 13A,B replacing CMcC with Q, -CMcC = -Q = r pi Cd/ (8Cb) = d/ds ln(V). This integrates to

V = Vz exp( -Q S) = Vz exp( - r pi Cd/ (8Cb) S) (14)

where S is range, r is air density, Cd is drag coefficient, Cb is the ballistic coefficient = mass/diameter sq. In equation 14 the velocity profile method completely overlaps the reference drag function Cd and ballistic coefficient multiplier, Cb. In fact, eq. 14 is one way to calculate Cd as a function of velocity.

Clearly Eq. 14 is not the same as Eq. 1, but by expanding the exponential in a Taylor series, then equating powers of S and dropping terms in S^3 and higher we can find that

B = - Vz Q = - Vz (r pi Cd/ (8Cb)) (15A)

A = .5 Vz Q^2 = .5 Vz (r pi Cd/ (8Cb))^2 (15B)

Looking at profiles for several hundred bullets, 15A is consistent with over 90% of the data, but 15B is not. A better approximation is

A = Vz Q^2, or more accurately, .575 * 15B (15C)

15A,B are accurate only as the range approaches zero. Note that Cd here is a function of velocity and not a constant.

As statistics from a sampling of bullets it appears that (as of 20 Nov.) estimates of A and B based on the ballistic coefficient Cb that are within 15% are

**A = 3 Vz/ ( 1.e9 Cb^2) (15D)**

1/B = -3.1 Cb (15D)

Eq. 14 offers an interesting option to the transit time equation Eq. 3. Eq. 14 can be integrated to produce, using eq. 15A

T = -1/B (exp( QS ) -1) = -1/B (exp( - B S /Vz ) -1) (16)

Drop and The Point Mass Approximation

Drop and The Point Mass Approximation

The discussion here mostly presumes that a bullet is a point that plows air. In reality it is a projectile with multiple axes for both motion and air resistance. The point mass approximation presumes the same drag function for drop as experienced along the flight path. It is hard to see how this can be true for both a cannon ball and a Lapua Scenar. One way to correct for transverse drag is to define a ratio of the transverse drag to the longitudinal drag, and evaluate the finite difference equations for vertical motion as in another section. This is a different, though similar, method from the adjustment noted above in the drag calculation, and implementation requires more calculation than is appropriate here. Whatever the method, a correction for the properties of longitudinal versus transverse drag should be considered when the drag coefficient is calculated directly from velocity profiles

**.**

**Explicit Fit to Three Points**

Least squares regression is beyond the scope of this note. For anyone who wants to construct their own spread sheet, here is a three point fit for the constants A and B. Assuming three points on the velocity profile (Vz, V1, and V2, separated by equal distances DS,

A = (Vz + V2 - 2*V1)/(2* DS^2) 17A

B = - (3*Vz + V2 - 4* V1)/(2*DS) 17B

If you set up a spread sheet, you can adjust these values of A and B and Vz to minimize the sum of squares of the difference between the calculated values and the measurements. This is the "hands on" equivalent of the least sum of squares regression.

**Notes and References**

1 The mathematical term integration -- construction of an integral -- means to find the area under a curve and express that area as a function of the position on the curve. When this is done using numbers instead of functions it is called quadrature. In the past to reduce the amount of calculation needed (remember it was once done all by hand) algorithms like "Runge-Kutta" were used. Today for most applications, and for rifle bullets in particular, we decrease the distance between samples along the trajectory until the change from one reduction to the next has no significant effect. Someplace around 1/4 foot sampling is a good place to start. Differentiation, meaning to find the derivative -- means to find the slope of a curve. The derivative of an integral leaves a function unchanged. The integral of a derivative may differ from original by a constant.

2 McCoy, Robert L; Modern Exterior Ballistics, Schiffer Publishing Ltd. 2012.

3 Pejsa, Arthur J; New Exact Small Arms Ballistics, Catalyst Graphics, Saint Paul, Minnesota, 2008

4 Litz, Brian; Accuracy and Precision for Long Range Shooting; Applied Ballistics LLP, 1507 Hanna Ave NE, Cedar Springs, MI 49319; 2012

The pictures used here are first, New Hampshire's Mt. Washington as seen from Mt. Jackson, and then the Presidential ridge as seen from the Sugarloafs, and a view of Mts. Tecumseh, Dickey, and Welch over Campton Pond.

**Acknowledgements**

Early in this project www.frfrogspad.com was inspirational and the good father there was most helpful in directing me to Jim Ristow at www.shootingsoftware.com, who was in turn most helpful in identifying several major issues in the velocity profile approach. Also early on, Mike Pomerantz mentioned that we live near several shooting places, and that I should think about it. Bruce Jorgenson particularly, and Don Johnson and Dr. Taoka have been very helpful in getting things running.

14 Nov 2013; 5 Dec, 20 Dec, 22 Dec., 17Jan2014, 12 April2014

Allan Ames