Mathematics of artillery

Summary: Mathematics is essential to the design and firing of artillery pieces.

Mathematics has had numerous military applications, including the development of artillery pieces after the invention of gunpowder in China in the fourteenth century. Mathematical formulas and calculations are critical to the design and use of artillery. The science of ballistics, which relies on mathematical formulas to study the flight paths of projectiles, also plays a major role in artillery development.

Engineer Benjamin Robbins invented the ballistic pendulum and is referred to by some as the founder of modern ballistics. Technological and scientific developments resulted in the modern use of artillery firing tables and computer-based firing calculation programs. Mathematics also plays a significant role in the ability to centralize fire control command centers and the use of indirect fire in which targets are not visible through a weapon’s sightlines.

Many mathematicians have worked in places such as the Ballistics Research Laboratory at Aberdeen Proving Ground, such as Gilbert Bliss, who worked on firing tables for artillery.

Early artillery weapons relied on mechanical energy to fire projectiles and were not of uniform design—making them large, cumbersome, and inaccurate. Technological innovations in the eighteenth and nineteenth centuries led to the development of standardized artillery with increased accuracy and mobility.

In the late eighteenth century, British Royal Artillery Lieutenant Henry Shrapnel created a shell (container) that held multiple musket balls and a time fuse that allowed the shell to travel longer distances before exploding, increasing the cannon’s range. High-explosive fragmentation shells and improved conventional munitions replaced shrapnel shells beginning in the early twentieth century.

Military scientists, weapon and projectile engineers, and soldiers have utilized the science of ballistics (the study of the flight of projectiles as they exit the weapon, travel through the air, and penetrate the target) since its early development in the fourteenth century to improve the accuracy and range of artillery.

Cannons, which first appeared in the early fourteenth century, spurred the development of ballistics. Early artillery crews used mathematics to determine the optimal angles at which to elevate their weapons for improved accuracy and range. Engineers also used mathematics to determine the angles at which to build fortifications to best defend against artillery bombardments.

Calculations of elevation, distance to target, weather conditions, projectile weight, and flight trajectory are necessary to achieve accuracy. Scientists and mathematicians, beginning with Italian mathematician Niccolo Franco Tartaglia, sought to improve the accuracy and reliability of early artillery pieces through ballistics. Tartaglia’s studies on a variety of cannons led to his determination that a 45-degree angle was ideal for firing—with the caveat that external factors such as air drag would affect the results. Tartaglia is also credited with the development of the first ballistics firing tables based on standardized weapons and projectiles.

Other notable mathematical advances in early ballistics included the theories of Galileo Galilei on the effects of the forces of gravity and air drag on the projectile’s velocity and flight path, as well as the parabolic nature of ballistic trajectories. In the early eighteenth century, English scientist Benjamin Robbins invented the Ballistic Pendulum, which allowed the measurement of a projectile’s velocity and the effects of air drag on that velocity. He also determined that air drag plays a much greater role in affecting a projectile’s velocity than gravity does. Sir Isaac Newton is credited with the development of formulas used to calculate aerodynamic drag, which he determined was proportional to air density, the projectile’s cross-sectional area, and the approximate square of the projectile’s velocity. However, Newton’s solution was incomplete, and mathematician Johann Bernoulli produced a more general solution. Mathematician Leonhard Euler integrated the various stages of a projectile’s flight to reduce the difficulty of the equations utilized in ballistics.

Artillery projectile designers use ballistics studies that calculate projectile properties, such as mass and diameter, based on the design specifications of the weapon in order to ensure the projectile will fit inside the barrel and generate enough energy to propel the projectile without damaging the weapon. Mathematical formulas are used to determine projectile design based on various input data including the force of aerodynamic drag, the ratio of the projectile’s velocity to that of the sound in the medium it will traverse, the properties of the medium, the projectile’s caliber (diameter), and the velocity at which it travels.

The mathematics of ballistics can be further broken down into internal, external, and terminal ballistics. Internal ballistics studies the flight properties of a projectile as it travels through the barrel of the weapon. A firing mechanism lights the gunpowder, which creates energy through the pressure generated by expanding gases. The energy is equal to the force times the barrel length. This energy forces the projectile through and out of the barrel. External ballistics studies the flight properties of a projectile as it travels through the air from the weapon to its intended target. Various formulas can be used to determine the kinetic energy of the projectile as it leaves the muzzle. Other calculations are then used to determine ballistic coefficient (a measure of a body’s ability to overcome air resistance in flight.). The distance and direction of artillery projectiles is affected by aerodynamic drag caused by a combination of air pressure (the disturbance of air around a projectile creating an area of low pressure behind it) and skin friction (the contact between the air and the projectile’s surface). Retardation is the measurement of the degree to which drag will slow a projectile’s flight speed and can be calculated by the following formula:

where R is retardation, D is drag, and M is the projectile’s mass. The ballistic coefficient is often used in place of drag because of the greater difficulty in calculating drag, which reduces along the flight path in relation to the decrease in velocity.

External ballistic formulas must also account for the fact that projectiles do not travel along straight flight paths. Physical and meteorological forces must be taken into account when determining or predicting a projectile’s flight path. These forces include yaw (caused when the nose of the bullet rotates away from a straight trajectory) and precession (caused when the bullet rotates around the center of mass). Terminal ballistics studies the impact of the projectile as it hits the target. Mathematical calculations can be used to study how a projectile’s design and flight features, such as velocity, shape, and mass, will affect its damage and wound capabilities.

Artillery firing requires the use of mathematical equations to determine range, elevation, and deflection, as well as the arc of fire and the probability of hitting the intended target. Artillery equation data also include the projectile’s initial velocity, which is further divided into vertical and horizontal velocity components. Calculating the distance a projectile travels is performed by multiplying the time the projectile is in the air by the velocity’s horizontal component. The needed angle to achieve a certain distance can then be determined by solving the equation for distance as a function of the angle.

Modern artillery crews rely on indirect fire, a technique developed in the early twentieth century in which a target is fired upon despite not being visible along sightlines. Indirect fire required more complex mathematical formulas and calculations, increasing the importance of specialized trained military personnel. These personnel calculated the range and bearing to the target. New techniques of determining the locations of enemy artillery batteries and subsequent firing data included flash spotting, sound ranging, air photography, and registration point. Indirect fire led to the development of graphical or tabular firing tables and the maintenance of a command center. Technological developments also allowed for greater adjustments to firing data based on such variable conditions as wind speed and weather. Initially, firing data derived from these tables was placed on the weapon’s sights.

Use of Computers

Battlefield computers began to appear by the 1960s and were in use by the British and U.S. military by the following decade. Computerized firing tables utilize input data to determine the angle and position of artillery, which weapons will fire, and how many rounds will be fired (although some military forces still rely on older instruments and human calculations as backups).

Firing data such as quadrant elevation, azimuth (an angular measurement in a spherical coordinate system), fuse setting, and projectile properties are inputted into the software program spreadsheets based on established data and standard conditions, which determine ideal firing information. The firing information is then corrected for deviations from standard conditions, such as meteorological conditions. Further technological developments include computer-based surveillance and target acquisition systems, global positioning systems, and laser rangefinders.

Bibliography

Carlucci, Donald, and Sidney Jacobson. Ballistics: Theory and Design of Guns and Ammunition. Boca Raton, FL: CRC Press, 2007.

Hackborn, William. “The Science of Ballistics: Mathematics Serving the Dark Side.” http://www.augustana.ualberta.ca/~hackw/mp480/exhibit/ballisticsMP480.pdf.

McMurran, Shawnee, et al. “The Impact of Ballistics on Mathematics.” Proceedings of the 16th ARL/USMA Technical Symposium, 2010. http://www.math.usma.edu/people/rickey/talks/08-10-25-Ballistics-ARL/08-10-23-BallisticsARL-pulished.pdf.