The first reliable range tables were compiled entirely by experimental firings. But these went only so far. Bear in mind that because of all the factors that affect trajectories, it's not enough to fire one shot at each elevation of interest. No, you must fire multiple shots, and average the values, to get reliable predictions. Multiply all this by the number of different guns, projectiles, and standard charges for those projectiles, and you can see that the ammunition expenditure would be quite considerable.
The old military term for a maximum range was a "random", representing apt skepticism as to the chance that such a shot would hit its target. French experiments of 1739–40 revealed that a 24-10, elevated to 45o, could range to 2250 toises (4793 yards)(Robins 234). In 1864, a 2.75" Whitwoth rifled gun achieved a range of 2,600 yards at 5o and 10,000 yards at 10o. (Bell 445; Newton 87). The effective range was of course much less.
Exterior Ballistics
Modern ballistics can be used to calculate the "first graze" (the initial point of impact of the projectile) and its impact velocity, given the muzzle velocity, the elevation of the barrel and a "drag function" for the projectile. These calculations couldn't be made accurately before the Ring of Fire (RoF) because of certain fundamental misunderstandings and fatal oversimplifications.
The leading down-time work on ballistics was Tartaglia's Nova Scientia (1537). The "physics" underlying Tartaglia's propositions is Aristotelian: a projectile is thought to follow first a straight line in which "impetus" is dominant, then a transitional curve, and then finally fall straight down ("natural motion"): "Wile E. Coyote" physics. Nonetheless, Tartaglia predicted that maximum range would be obtained if the projectile were fired at an elevation angle of 45o—true if the trajectory is in a vacuum.
Galileo has also been studying ballistics; unfortunately, he didn't publish his work (Discourses and Mathematical Demonstrations Relating to Two New Sciences) until 1638. He was the first to point out that gravity wouldn't affect the horizontal motion of the projectile, that a body without an initial upward motion would fall a distance proportional to the square of the time elapsed, and that the combination of these propositions indicated that the path of a projectile would be a parabola. This is all true—in a vacuum.
The range in a vacuum is easily calculated:
Rvac = V2 sin (2*theta) /g
where V is muzzle velocity, g is gravitation acceleration and theta is the elevation angle. In a vacuum, the maximum range would be at an angle of 45o, and for any lesser range, there would be two equally acceptable elevation angles for achieving it.
Galileo's disciple Evangelista Torricelli published many ballistics theorems in Opera Geometrica (1644). When Giovanni Renieri complained that his experiments did not agree with Torricelli's formulae for the relationship of the point-blank range to the maximum range, Torricelli reminded him that the text was intended for philosophers, not gunners.
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Because of air resistance, the maximum range is lower than Rvac and it's achieved at an elevation angle that's less than 45 and may be as low as 30o. (Rinker 332; Douglas 43, 36). The lower the elevation angle (thus, the flatter and shorter the trajectory), the less the effect of air resistance. High angle trajectories nonetheless had their uses, mostly in attacks on fortifications where you needed to put a shell over a wall to hit a higher value target beyond.
In 1668–69, Christaan Huygens demonstrated that water resistance was proportional to the square of the speed of the object moving through it (true for low speeds), and inferred that the same was true of air resistance. Newton later advanced a similar proposition in his Principia.
The air resistance (drag force) is
0.5 * rho * CD * A * V2
where rho is the density of air (which changes with altitude, CD is the dimensionless drag coefficient, A the reference area for which CD is determined (typically the frontal area for a projectile), and V the air speed. CD is a function of projectile shape and, unfortunately, speed.
At typical muzzle velocities, the drag force is more than twenty times as strong as the gravitational force.
The equations of motion for a projectile subject to both gravity and air resistance are easy to write, given calculus and knowledge of the resistance as function of speed, but difficult to solve. Some first-class mathematicians addressed this problem, including Bernoulli and Euler. Their work is touched upon by EB11/Ballistics, which goes on to discuss the Siacci-Ingalls approximation method. That was still in use in World War I. EB11/Ballistics also describes in some detail how Bashforth constructed his ballistic tables.
It is likely to occur to the mathematicians in Grantville that the equations can be "solved" by numerical integration (calculating position, velocity and acceleration in small time increments, say a millisecond at a time), and indeed that is the only viable approach if the speeds are supersonic and the trajectories are high (so air density varies with altitude). But even with simpler trajectories, it may seem simpler in Grantville's "computer culture" to use numerical integration rather than reconstruct the complex analytic approximations of Bernoulli, Euler and Siacci-Ingalls. (cp. Cline).
Doing this requires knowing the dependence of air density on air temperature and pressure, and for high trajectories of those on altitude (the aviators in Grantville should have this information) . Also, one must determine the drag function (the speed dependence of the drag coefficient) for the projectile of interest, which our characters can do by quantifying (with ballistic pendulum), for each of a variety of charges, its muzzle and down-range velocity from a particular gun.
Just to show that it can be done, I have constructed an Excel spreadsheet that uses the Runge-Kutta fourth order approximation method (which should be described in a standard textbook on numerical analysis, and thus in the personal library of one of Grantville's mathematicians) to solve the linked differential equations of ballistics. This can be used to construct a range table, if we know the drag coefficient as a function of speed for the projectile, and the muzzle velocity with which the gun projects it. And for that, we need to be able to measure projectile speed.
Projectile Speed Measurement
The real breakthrough in exterior ballistics was the invention (1742) of the ballistic pendulum by Benjamin Robins, who used it to measure the speed of projectiles at the muzzle and at various ranges (the latter permitting the effect of air resistance to be quantified—Douglas 129). This device is described in EB11/Chronograph, but in essence the projectile strikes a pendulum and transfers its momentum to it, causing it to swing.
Robins not only confirmed that the normal drag was proportional to the square of the speed, he detected the sharp (perhaps three-fold) increase in resistance at, he reported, 1100–1200 fps, that became known later as the "sound barrier."
Modern shooters have made their own ballistic pendulums with what appears to be reasonable accuracy (main worries are projectile deformation and deflection, friction, gravity and calibration), so this can definitely be done in the new time line. The accuracy of the ballistic pendulum is a respectable 2% (Rinker 148).
The catch is that cannon balls are heavier than bullets, and the ballistic pendulum must be scaled up to match. Hutton used an 1800 pound pendulum for studies on 6 pound balls, and in 1839–40, to measure the speed of 50 pound projectiles, a six ton receiver was employed. (Bashforth 25).
Reverend Bashforth achieved even greater accuracy by timing when the projectile passed through wire screens separated by a known distance; the penetrations interrupted the electric current to a chronograph. (Cantwell 46). Gun chronographs are described in EB11/Chronograph.
Drag Coefficient
The drag coefficient is dependent on the Reynolds number, which is proportional to both the length and the air speed of the projectile. It's also dependent on the Mach number, which is the speed as a fraction of the speed of sound (340.45 meters/second; 1117 feet/second; sea level, 15oC, 59oF). For typical projectile speeds, Mach number is more important.
A projecti
le traveling at a speed close to the speed of sound (Mach 1.0) exhibits a mixture of subsonic (under speed of sound) and supersonic (over) air flows. This range of speeds at which this mixed flow occurs is called transonic. A projectile is said to be in the subsonic regime if it is traveling at less than Mach 0.8 (some authorities would say 0.6 or 0.7), in the transonic regime at Mach 0.8–1.2, and in the supersonic regime at over Mach 1.2.
At subsonic speeds, drag is primarily frictional drag (air retarded as it passes over the surface) and pressure drag (air pushed out of the way). The lower end of the transonic range (critical Mach number) is where wave drag (air compressed) first appears.
The drag coefficient varies depending on the shape of the projectile. It should be mentioned that since Grantville is in rural West Virginia, it is going to have a higher-than-national average of hunters, of firearms (probably more firearms than people), and of books and software relating to firearms, including ballistics. They may thus be able to construct a reasonable drag function without experimentation, at least for shapes similar to the standard G1–G7 shapes.
Finding a drag-speed function for a cannon ball might be tricky. While your characters will have to determine it the hard way, you as an author may look it up. Douglas (132) gives a table of air resistance to a cannon ball at velocities of 100–2000 fps (cp. Guilmartin 296; Allsop 120).
Here are some range data that I calculated with my spreadsheet:
By way of comparison, in a 1796 Admiralty test, a 24-pounder elevated 2o achieved 1274 yards with a one-third charge and 992 yards with a one-fourth charge of cylinder (Red LG) powder, and 1020 with one-third charge of the old Blue LG powder. (Gardiner 129).
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Historically, experimental firings of a small number of standard projectiles were used to construct retardation (deceleration) functions (retardation=kVm), which in turn allowed approximate calculation of the trajectories for a wide variety of elevations and each service gun's expected muzzle velocities (for each of its standard loads), under "standard conditions". (It was more accurate overall to use the average muzzle velocity for the service life of the gun, rather than the "new gun" design velocity.)
Based on Krupp's experimental firings in 1881, Mayevski and Zaboudski formulated a seven-piece drag function for a "standard projectile", with e.g. m=2 (corresponding to a constant drag coefficient) for speeds below 790 fps, m=5 for 970–1230 fps, and m=1.55 for 2600–3600 fps. (Ingalls iv). The British developed a somewhat different drag function in which m was as high as 6.45.
Ballistics tables were developed to reduce the work involved in calculating range tables (Hackborn). The early-twentieth-century tables pre-computed, for various velocities, the space, altitude, inclination and time functions used by Siacci's method for calculating the trajectory. The horizontal and vertical coordinates of the trajectory were dependent on these functions, and on a properly corrected "ballistic coefficient" (unity for standard projectile under standard conditions).
The ballisticians assumed that the retardation functions could be applied to a non-standard projectile by adjusting a parameter known as the "uncorrected" ballistic coefficient, which in turn considered projectile weight, diameter and "coefficient of form," the last essentially embracing not just form but all other projectile characteristics that would affect its trajectory. The coefficient of form for each non-standard projectile was itself determined from a more limited set of test firings. The actual drag at a given velocity was assumed to be the standard drag divided by the ballistic coefficient, corrected for non-standard air density, etc.
The ballisticians erred in assuming that the coefficient of form was a constant, i.e., projectile shape would have the same effect at all velocities. You could, of course, construct a separate drag function for every projectile, and throw the ballistic coefficient "out the window."
In practice, the ballisticians developed a small number of retardation functions, one for each "family" of projectile shapes, and accepted the residual imperfection of the coefficient of form.
Ricochet Fire
This is the reflection of a shot by a surface. EB11/Ricochet says that ricochet fire was first employed by Vauban at the siege of Ath (1697). This is poppycock. Shakespeare refers to the "bullet's crazing" (grazing), causing it to "break out into a second course of mischief, killing in relapse of mortality.: Henry V, Act IV, scene iii. And Bourne, The Art of Shooting in Great Ordnaunce (1587), has a section entitled, "How and by what order the shot doth graze or glaunce upon lande or water," which recognizes that the ricochet occurs only if the angle of incidence is shallow. Nonetheless, I suspect that it was not common in the seventeenth century to deliberately elevate the gun so that the shot would strike an enemy ship by ricochet. A particular advantage of ricochet fire was that it tended to strike the target near the waterline.
Ricochet is possible only if the shot strikes water at an angle less than the critical angle; Douglas (108) advises that the angle of incidence shouldn't be greater than 3–4o, while Beauchant (30) favors under 2o. The critical angle depends on the projectile; Birkhoff (1944) proposed that it is 18o divided by the square root of the specific gravity of the projectile (Johnson); for an iron cannonball, that's 6.36o (cp. Adam 111). The angle of incidence depends on the angle of elevation, the height of the muzzle above the water, air resistance, and wave action. Ricochet is more effective when the water is smoother, and firing to windward (lee sides of waves are steeper). There is of course some loss of energy at each bounce, so at least a one-third charge of powder is desirable if you're shooting at a large ship. (Beauchant 31). Ricochet can't be used effectively with rifled projectiles; because of their spin, they are reflected at a high angle. (Scott 25).
It's really amazing how much ricocheting can increase range. By way of an extreme example, the Vesuvius fired (1797) a 43.7 pound shot at a 1o elevation; its first graze was at 358 yards; but it ricocheted a total of 15 times, achieving an extreme range of 1843 yards. (Douglas 209). The following table relates elevation, charge, distance to first graze, and extreme range:
(Beauchant 20, 26)
There are empirical formulae of uncertain reliability for estimating ricochet range (Abbot59ff) but I doubt they are in Grantville Literature.
In WW II, the Fifth Air Force experimented with "skip bombing"—essentially, a low-altitude, high-speed approach so the bombs would ricochet toward the enemy. However, they abandoned this tactic in favor of "masthead height" bombing: "to eliminate the need to calculate the ricochet distance, they timed the release to hit the side of the ships, instead of bouncing short." (Gann 15).
Maximum effective range (MER)
Exterior ballistics can identify the range at which a particular projectile, fired at a particular muzzle velocity and elevation, can strike with a particular impact velocity. And that in turn may be compared with the empirical formulae of terminal ballistics (part 5) to determine how many inches of wood or iron it will penetrate.
What it can't do is determine the likelihood that the projectile will strike the target, and is that probability high enough to warrant firing. Is the projectile expensive and available only in small numbers (like a torpedo or missile), so you must make the shot count, or cheap and plentiful?
Accuracy, Precision and Trueness
If there is a single trajectory by which the gun, fired at a particular instant, will hit the point of aim, then it follows that there are essentially three kinds of firing problems that result in a different point of impact: errors in traverse, elevation and muzzle velocity. Errors in traverse result in a lateral error at the range of the target; errors in elevation result in a vertical error (if aiming at a "vertical target" like the side of a ship) or a range error (if aiming at a "horizontal" target like the deck). Errors in muzzle velocity cause vertical and range errors directly, and lateral errors indirectly (by making too much or too little allowance for wind deflection or ship motion during the time of flight).
There are two ways of coping with errors: minimizat
ion and compensation. We can minimize variation in powder strength, shot size, etc. There are some errors that we can't avoid, but if they are of predictable magnitude and direction, we can compensate for them. For example, if spin causes a rifled projectile to drift 10 yards left at a range of 1000 yards, we can aim enough to the right of the target so it will drift left onto it. Likewise, we can compensate for known wind, gun platform motion, and target motion.
Precision measures how closely the impacts are grouped together; accuracy, whether they hit the target. It's also helpful to recognize a concept that used to also be called accuracy but which is now (ISO 5725 for the measurement community) called "trueness": the distance of the mean point of impact to the point of aim (the desired point of impact).
It's possible to measure the precision of a gun (although this assumes that you have minimized variation in ammunition, elevation and atmospheric conditions) but a gun doesn't have an inherent accuracy. Any reference here to the "accuracy" of a gun means of that gun operated by a typical trained gun crew.
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"Accuracy" is most often casually stated as the number of hits made on a target at a particular range; hopefully, the records will also state the dimensions of the target.
The most useful method of quantifying either accuracy or precision is in terms of the mean error, laterally and vertically (or in range), of the points of impact from the point of aim (for accuracy) or from the mean point of impact (for precision), for a given range. One may also present the mean absolute error, the mean radial distance of the points of impact from the reference point. Unfortunately, most mean error data is from the mid-nineteenth century or later.