Long-Range Ballistics
By Steve Reichert
Interest in long-range ballistics started shortly after the invention of guns in the
tenth and eleventh centuries. Cannons rather quickly replaced catapults and other
siege weapons used to start fires and knock down walls. It is known that Leonardo
da Vinci provided some level of trajectory computation to his benefactor, Ludovico
Sforza, the duke of Milan, around 1500.
The application of Newton’s Laws in the form of vector calculus modeling of ballistic
trajectories in the late 1800’s and early 1900’s was a great leap forward. However,
it quickly became apparent that exact solutions to the equations were out of reach
due to the non-linear compressibility properties of air. The works of the Italian
mathematician Francesco Siacci in the late 1800’s and American mathematician and
ballistician James Ingalls in the late 1800’s and early 1900’s simplified the
compressibility problem by providing the first measurements and tabulation of
ballistic drag. Solutions to the trajectory equation, even in one dimension, remained
a very laborious task.
It is edifying to note how the shot range of interest has evolved over historical time.
In the time of Punic Wars, the range of a Roman ballista or trebuchet was a few
hundred yards and nobody worried about such complexities as wind. By the time of
Galileo or even Leonardo, this had not changed much. Certainly by the time of the
American War of Independence there were marksmen who were interested in
shooting to ranges of three hundred to even five hundred yards, though the means
of computing such shots did not yet exist. World War II was the first time precomputed trajectories (as opposed to log-book entries) were widely used, but they
were restricted to artillery. Snipers and other marksmen were still using logbooks
and experience to guide their shots, limiting them to around five hundred to
extremes of one thousand yards. This continued into and beyond the Vietnam War.
The proliferation of computers in the 1970s and 80s allowed exploration of solving
simplified ballistic equations to increase accuracy for small arms fire. At that time,
the range of interest was at least partly limited by the quality of barrel manufacture
and available long-range, gun mountable optics, so ballisticians still limited their
computations to about one thousand yards. Modern wars, after 1990, have
demanded a continuous extension of the ability to accurately deliver a rifle bullet to
longer and longer ranges. As present military sniper kill ranges attest, the present
range of interest is well in excess of two thousand meters and there is some interest
in the possibility of reaching three thousand meters.
One of the first problems solved by the new electronic computers (at that time,
using vacuum tubes) was calculation of ballistic trajectories for artillery guns during
World War II. These computations were done on the very first general purpose
computer, the ENIAC at the University of Pennsylvania.
The first really complete publication of ballistic modeling and solution techniques
was “Modern Exterior Ballistics”, by Robert McCoy of the US Army’s Ballistic
Research Laboratory. This book has become the “Bible” of technical ballistic
trajectory theory and practice.
There are significant sources of error in the presenting the modeling and standard
mechanisms used for solving the ballistic trajectory equations. The most important
of these is the simplification of the problem by leaving out all modeling of the
rotational dynamics of the projectile.
The full ballistic trajectory equation suite consists of six (6) coupled second order
differential equations. Each of these equations represents the dynamics of one
“degree of freedom” of the bullet. The enumerated degrees of freedom for a bullet
are:
1.
2.
3.
4.
5.
6.
Down-range position
Horizontal cross-range (windage) position
Vertical cross-range (height above like of sight) position
Projectile roll (rifling twist) angle
Projectile pitch angle
Projectile yaw angle
The first three of these are the positional equations, while the second three are the
rotational dynamical equations (also known as the “attitude” equations). Each of
the equations has a number of forcing function terms that cause some part of the
dynamics for that particular degree of freedom. Gravity, for example, is a forcing
function term, as are the drag forces (including wind).
The coupling of the equations means that there are forcing function terms in each of
the equations that depend on the current solutions of one or more of the other
equations in the set. For example, the instantaneous drag on the projectile along the
down-range direction depends on the instantaneous projectile pitch and yaw angles
while the time rate of change of the pitch and yaw angles depend on the down-range
velocity, which, in turn, depends on the summed instantaneous drag. The solution
of the entire set of fully coupled equations is a daunting task, indeed.
From the standpoint of the shooter, projectile rotational dynamics is not particularly
interesting. The shooter wants to get the projectile to the target, which has
everything to do with its position. It is true that there are attitude terms in the
positional equations, but their effects are rather small compared to the other forcing
terms like gravity, drag and wind. Leaving out the computations of the rotational
dynamics does not have a huge effect (but it is not insignificant, either) on the
position solution at closer ranges and considerably reduces the complexity of
actually getting a solution. The most obvious effect, but not the only effect, of this
simplification is that the force term that results in spin drift is no longer being
computed. Modeling spin drift without the attitude terms requires an after-solution
add-on using some form of heuristic approximation. Different products use
different, usually proprietary heuristics for this purpose or simply ignore the spin
drift altogether. For a standard 168 grain .308 bullet with a muzzle velocity of about
2650 feet per second and a right-hand twist of one turn in 11.75 inches, the spin
drift results in about a 6.7 inch position offset to the right1 at one thousand yards
range. Dropping the angle equations reduces the mathematical model to what is
known as a point mass model.
Another source of error in the present solution methods is due to the “flat fire”
approximation. The item of highest interest to the shooter is the vertical angle that
the bullet must have as it exits the muzzle (i.e., the vertical scope setting). The
method used to obtain this angle is to assume that the shot is taken flat, that is, with
the barrel completely horizontal. The method assumes that there is no initial angle
on the gun and computes the trajectory out to the target range as if the Earth would
not stop the bullet if it hit the ground. The projectile will be pulled down by gravity
during its trip to the target and the solution to the ballistic equations will result in a
number representing how far down it got pulled. Simply computing the arc-tangent
of this number divided by the range will result in the required angle… but not quite.
The flat fire solution presumes, by its very nature, that the initial down-range
velocity is the muzzle velocity. However, when we raise the gun by the angle
specified by the flat fire solution, it results in both an initial down-range velocity and
an initial upward velocity. These two velocities have a Pythagorean relationship to
the muzzle velocity. The result is a small reduction in the down-range initial
velocity, so the bullet actually takes a little longer to reach the target than the flat
fire approximation predicts, so it falls a little farther and the shot arrives a little low.
Other sources of inaccuracy are encountered when shooting at long ranges on
inclines (up or down). The first is that the gravity vector is no longer aligned with
the vertical axis. This puts some gravitational acceleration into the down-range axis.
The result is that there is either more or less effective drag on the projectile,
depending on whether the shot is going up- or down-hill. The second is much more
subtle. For a long-range shot on a significant up- or down-hill angle, there is an
associated change in actual, Earth oriented altitude. This altitude change results in
changes in density, pressure and temperature along the trajectory, and associated
changes in Mach number and drag.
All of the sources of imprecision discussed above are very small to non-existent at
short ranges. However, the physical processes and time/distance evolution of the
effects are non-linear and grow to considerable significance at extreme long range.
There are other causes of error in addition to those mentioned above. They are
much more mathematically and/or physically obscure and go beyond the scope of
the current document, but they are no less significant than those discussed here.
The common point mass approximations that, per force, ignore attitude force terms
and subtle geometric and atmospheric effects simply cannot accurately compute
precise solutions beyond about eight hundred meters range. This is the reason
ballistic calculators resort to “truing”, or correcting for errors occurring in the first
(or second…) round shot. Getting past these limitations requires an entirely new
approach to solving the ballistic problem.
Note that this is only the offset due to the spin drift. There is another 2.5 inches of
right offset in the northern hemisphere due to the horizontal component of Coriolis
effect at this range and time of flight.