# Ballistics

A gun requires a sighting mechanism if it is to be able to hit its target, but you cannot construct such a device before you have a complete understanding of how the weapon will function when firing under different conditions.

## Contents

## Ballistics

**Ballistics** is the physical science that governs how a projectile is fired and how it flies.^{[1]} Ballistics for a powerful gun are very involved due to the number of variables that effect where the projectile goes:

- changing the gun's elevation will affect the range and also the shell's time-of-flight
- the shell's spin as it leaves the rifled barrel induces a lateral wander called *ballistic drift*
- variations in powder temperature will affect the muzzle velocity
- the ambient temperature and humidity will affect the amount of *drag* the shell experiences
- the gun's muzzle velocity drops with successive firings, as the bore wears out

The number of variables and the range each might vary over are considerable, and the expense of conducting test shots is such that not enough can be executed to provide the full understanding needed to make the required gunsight apparatus. Ballistic science was used to help fatten the data obtained in a limited series of test firings into a voluminous description of how the weapon would perform under the many conditions *never actually tested*. This synthetic dataset was called the weapon's "range table", and it served as the basis for the design of gunsights and fire control equipment that would help it hit home.

## Range Tables

### Format and Content

The most desirable form for such documents was generally a header and a lengthy tabular data set and possibly a few ancillary tables.

The header indicated the gun, shell and powder charge (the combination of which I call a "gun system", although there seems to have been no contemporary term) whose joint operation was being described, as well as specifics on the muzzle velocity obtained and the atmospheric conditions for which the firing data corresponded.

The tabular data detailed the performance of the weapon system throughout the entire envelope of ranges it it could attain on mountings it would be fitted to, sorted from minimum to maximum. The first column indicated ranges (in increments of 100 yards or meters, depending on the nation), and the remaining columns the other parameters such as the angle of elevation required to loft the shell to the range, the time-of-flight, the terminal velocity at impact (less than muzzle velocity, owing to drag), and the angle at which the shell would fall. Of these, the range/elevation relationship in the first two columns was most important.

Additional tables, when provided, might include other information, such as

- the amount of lateral deviation at each range attributable to ballistic drift (an aerodynamic effect of the spinning shells) - the "50% errors" along and across the line of fire at each range (a statistical measure of the the weapon system's precision) - the rate at which muzzle velocity would be reduced after a given number of rounds had been fired.

Such data was essential to create equipment and procedures to support the gun's use along its entire service life.

### Proof Firing a Gun System

The Royal Navy had a large firing range at Shoeburyness where new weapons could be trundled out for firing. Presumably, no one lived on the reservation (or only those homeowners possessing both a sensitivity to price and a contrarian's take on actuarial science).

One might imagine that a sample gun might just be fired so many times at a number of elevations that the range table could simply be the result of those firings where the shell traveled an even multiple of 100 yards (or meters). But such an ideal method would be both very time consuming and expensive when one considers the cost of shells, powder and that the guns wear out quite quickly and each successive firing occurs at a progressively degraded muzzle velocity. These factors dictated a different approach be taken.

A ranging and accuracy "practice" was conducted according to a program crafted by the Ordnance Board and resulted in a report that went something like this. The background conditions were first reported, and contained the following information:

* date of practice * place and direction of firing (e.g., "along line of pegs 'G' bearing N. 80 E.") * gun type and serial number * complete firing history of the gun (e.g., "rounds previously fired, 3 proof and 2 full charges") * mounting used * identification of type of projectiles used * powder charge employed * temperature of charge * barometric pressure * air temperature by dry bulb and wet bulb measures

The test firings would occur in approximately 4 distinct "series" of firings, each series comprised of 5-10 shots all taken at the same quadrant angle of elevation. For the first series (at the lowest elevation), there were be additional means taken to discover the muzzle velocity and the angle of "jump" of the gun.

Muzzle velocity was not directly measured, but back-solved for by measuring the shell's velocity a short distance from the gun. This velocity measurement was obtained by having the shell pass through two skeins of electrified wires around 100 yards downrange. As it broke these wires, high speed timing devices measured the interval between the breaking of the two separate circuits. Some mathemagic allowed the small loss in muzzle velocity to that point to be factored out. "Jump" (a small vertical "flinch" a mounted gun gives as it discharges its shell) was measured by having the projectile pass through a screen some 400 feet from the muzzle, and comparing the height of this hole from a straight line down the bore of the gun before it was fired, possibly correcting also for the short distance the shell would have already fallen at that 400 foot distance.

Additional data recorded for each shot taken were time of flight to impact (in 100ths of seconds), range attained (in yards), lateral deviation (in 10ths of yards), velocity at the skeins (in feet per second) and the distance these were from the muzzle, muzzle velocity calculated from that figure, wind speed in feet per second, wind direction in compass degrees, and apparent "jump" of the gun at firing, in arc minutes, by judging where the shell holed the screen.

Once the data was collected, the individual shots were grouped by series, and mean values for the shots within each series computed. Additional data in this reduced table also showed that the process was mindful of how the muzzle of the long gun was itself elevated "above the sands" as the gun elevated through its arc, and the wind was helpfully decomposed into components along (head or following?) and across (right to left or left to right?) the line of fire and by some means computed at a height of 101 feet though there is no direct indication of the height at which it was measured. The last thing in the series table were the measures of the variety in range and lateral deviation observed across the shots within the series. This last was to serve as the basis for computing 50% errors.

### Analysis of Firing

Once the above data was in hand, the considerable task of massaging the numbers began. The work focused first on neutralizing the conditions or each firing to a common, standard set of conditions. Fine points tackled here included correcting "muzzle height above sands" to factor out the curvature of the earth, calculating the "angle of depression" (that extra amount of "looking down" necessary to see the impact point in its "hole" behind the curving earth), and considering "droop" of the barrel under its own weight and imperfect rigidity. Approximations had to be employed to factor out the slight loss in muzzle velocity from one shot to the next, even in a small proof firing exercise. The "muzzle height above sands" had to be tweaked to factor out the curvature of the earth between firing position and impact position. The tendency of guns to "jump" slightly upon firing caused the elevation at the muzzle to increase slightly as the shell left. This was computed. Each shot finally had a "departure velocity" computed, which wound up as:

departureElevation = quadrantElevation + droop + jump

Wind was factored out and even considered to have had a small effect in adding to "jump", according to "Greenhill's treatment". For each shot, then,

jump += arcsine( sine(angleOfDeparture) * windFollowing / muzzleVelocity)range += windFollowing * timeOfFlightmuzzleVelocity -= windFollowing

After each shot's various individuals differences were neutralized, an arduous process was undertaken to derive a "coefficient of reduction" for each firing series. This procedure, as documented in Manual of Gunnery for HM Fleet, Volume II (1917), alludes to the O. B. Ballistic Tables which I do not have, so I'll skip on that part for now. Suffice it to say that each series has a coefficient of reduction computed for it, and a suitable single value chosen somewhere within the set is then selected to characterize the entire weapon system. This coefficient of reduction served the considerable function of describing how the gun system differed from an idealized model. As tenuous as this method seems to me, the Admiralty placed great confidence in these approximations, at least up to elevations of 15 degrees^{[2]},

The range table was then created, but only around 5 or 6 of the hundreds (200, in the case of a Royal Navy table extending to 20,000 yards) of rows were actually computed. Rather, 5 or 6 rows scattered along the table (say 1.5, 3, 6, 9, 12, and 15 degrees angle of departure) would be computed from the O. B. using the coefficient of reduction, and all further points then obtained by graphical interpolation of these 6 points, presumably, using a draughtsman's spline.

While I am confounded by my lack of insight into the O.B. Ballistic Tables and the computation, meaning and use of the coefficient of reduction, I have to express considerable surprise that the canonical documents of this time actually contain data so indirectly based upon a tiny number of experimental trials. While clearly there would be some difficulty in measuring angle of descent and remaining velocity of shells, the other vital relationships of range to elevation (or drift, or time of flight) would seem better sourced from the firing trials themselves, and not to distortions of a single model of projectile flight.

## Appendix: Ballistic Drift

This section will be unduly long, as it touches on my own attempts to understand the magnitude and nature of this force which generally causes shells to "drift off" to the side as they fly downrange. Like any regular phenomenon influencing the fall of shot, it behooved a nation to develop and understand of drift's nature and to create mechanisms that might factor it out of the gunner's problem before he even gets called to action stations.

It is not easy to find comprehensible information on ballistic drift as a *physical* phenomenon. Indeed, it is hard to find specific data on a given weapon system for how far the shell would deviate laterally (generally to the right, given the customary direction of twist in rifling) as it traveled to various ranges. However, the 1918 Range Tables offer a little explanation for how the Royal Navy treated drift and some of the compiled range tables within include fairly complete drift data, although one might be suspicious as to which are experimentally obtained and which are the result of a mathematical model for drift.

It was observed that the length of the projectile was a primary factor for why different shells exhibited different drift behaviors^{[3]}, and a mathematical description of drift angles was derived for use in the form:

driftAngle = someConstantForThisGunAndProjectile * tangent(elevation)^{[4]}

However, not all range tables within the volume have drift expressed in this manner, as determination of the constant for all guns was not possible before going to press.^{[5]} The formula is puzzling, as it seems as though not only was a rule of thumb being entrusted to model drift for the world's pre-eminent navy, but that it had been recently conceived and relied upon empirical constants still being sorted out.

Critical thought indicates that the formula might not express the drift angles observed in test firings, but those allowances for drift provided by the Royal Navy's oft-employed mechanical drift correction built into many of her gun sight apparatuses wherein the sightsetting gear and scopes were simply cocked to the side a smidge so that as the gun was elevated a deflection was progressively induced. It is not clear, then, whether the drift data in the volume are results taken from this formula and why range tables did not include drift data based on a few test firings at different ranges and with other values plugged in after interpolating their likely best values.

Ignoring for the moment our doubts as to where these data came from, a graphical analysis of the drift data from one range table (Table 50) does not seem to imply that it was spat out by the RN's drift angle formula. One caveat is that the first few data are likely distorted due to rounding errors, as the first 5 appear with only a single significant digit. This first scatter plot explores the correlation between the RN's formula and the data in the table.

The correlation seems fair, but a visual inspection clearly indicates a relatively weak relationship at the short range shots. Let us ignore the Royal Navy's best published thinking on the matter and work with a hypothesis that the lateral deviation observed in practice was the result of a constant force acting upon the shell throughout its flight, and that maybe the data in this one table are derived from actual test firings (possibly with some interpolation). This seems like a reach, but proves a gamble well worth taking: we achieve an *even better fit*, even when including the first 11 data points with their presumed rounding errors!

The conclusion is that *extremely* faithful simulation of drift can be attained by taking the most extreme (range, time-of-flight, drift) datum available for a gun/charge/projectile system and calculating a constant lateral force which, if applied continually during flight that produces the drift behavior observed in this single historical firing. A big win for the anal retentive game coder who lacks a PhD in Physics.

## Footnotes

- ↑ To be more precise, ballistics is treated as two separate realms: "Internal Ballistics" focuses on how the charge propels a projectile within the bore of a gun and "External Ballistics" focuses on how the projectile flies once it has left the gun. External ballistics is so vastly predominant in its import that one can generally assume when reading mention of "ballistics" that the writer means to refer to "external ballistics".
- ↑ "... it is unlikely that the results obtained at practice will differ widely from the range tables, and should a large error appear to exist it is generally advisable to seek the cause elsewhere than in the tables."
*Range Tables for HM Fleet, Volume I, 1918*, p. 5 - ↑
*Range Tables for HM Fleet, Volume I, 1918*, pp. 7-8 - ↑ Formula adopted
*vide*Gunnery Order No. 200/1916*Range Tables for HM Fleet, Volume I, 1918*, p. 8 - ↑
*Range Tables for HM Fleet, Volume I, 1918*, p. 8

## Bibliography