Ballistics is the science that studies the propulsion, motion, and destructive action of projectiles, chiefly missile weapons such as bullets, \Tartillery\t shells, \Lbomb\ls, rockets, guided missiles, and military satellites (see \Trockets and missiles\t and \Lsatellite\ls, \Tartificial\t). The theory and techniques of ballistics have broad applications to technology--including the formation of metal parts by explosion, and the development of cartridge-actuated devices for industry and of heat shields and aluminum bumpers for spacecraft. Ballistics is also applicable to scientific research in fields such as geophysics, geodesy, meteoritics, and planetary exploration. Development of Ballistic Theory The first systematic treatment of the ballistics of gunnery was given by the Italian Niccolo Fontana, better known to historians of science as \TTartaglia\t, in his Nuova scienzia, which was published in 1537. A professional military engineer, Tartaglia served as a consultant on scientific problems to the rulers of several principalities. The master of ordnance at the castle of Verona suggested that he consider what angular elevation of a gun barrel would yield the greatest range for a shot. Tartaglia found that an elevation of one-half of a right angle--in traditional measure, 45 deg--was the inclination required. The theory of exterior ballistics was rapidly developed early in the 18th century, after the principles of dynamics and the methods of the calculus had been established by Galileo, Newton, and Leibniz. But this important work in theory was largely an exercise in pure mathematics that had no immediate effect on practical gunnery, because no acceptably accurate way existed to measure the muzzle velocity of any firearm. Such a method was first suggested by the astronomer \TCassini\t in 1707; the instrument itself, the ballistic pendulum, was invented by the Englishman Benjamin Robins in 1740. In this device, a large pendulum bob is suspended from a tripod. A bullet of small mass is fired into the bob, causing a swing of large amplitude that can be carefully measured. The velocity of the striking bullet can be determined from the known masses of the bullet and bob and the amplitude of the swing, using the basic laws of physics. Drawing on his own experience and the writings of others as far back as Leonardo da Vinci, Robins suggested some useful directions for research and development in gun and projectile design. Noting that large quantities of gas escaped past round shot in the smooth-bore weapons of this time, he proposed the use of breech-loading weapons (loaded at the rear of the bore) with rifled barrels (containing spiral grooves), and elongated projectiles of close fit in the bore. He described these design proposals in his New Principles of Gunnery; they were gradually incorporated into ordnance-engineering practice during the 19th century. A breech-loading infantry rifle, the needle gun, so called because of its long, sharp firing pin, was invented by Johann Dreyse and issued to some Prussian regiments in 1841. A serviceable breech-loading artillery rifle was developed by Major Cavalli of Sardinia in 1845. Pointed cylindrical projectiles became standard issue for both small arms and artillery. Bullets were made of soft metal so they could be seated at the base of the rifling in small arms, and copper rotating bands were added near the base of artillery shells. By these means, the gases produced by the burning powder were retained behind the projectiles, which were induced to spin as their seating grooves were forced forward along the helical curve of the rifling. Interior Ballistics Interior ballistics is the study of the propulsion of projectiles by forces derived from the expansion of gases burning within a gun or rocket motor. Although the burning proceeds by similar stages in a gun and in a solid-propellant rocket, the pressures developed within a closed gun breech are much higher than those in a nozzled rocket motor; therefore, the metal parts of a gun system--the chamber, barrel, and recoil mechanism--are more complicated and must be much stronger than those of a nearly recoilless rocket launcher. In addition, motion is imparted in quite different ways to projectiles fired from guns and to missiles carrying both warheads and gas-reaction motors. Guns. During the early development of the interior ballistics of guns, the muzzle velocity and the maximum pressure of the gas in the chamber were considered the two most important physical variables to predict and measure. These quantities were found to be functions of given firing conditions, usually the known characteristics of projectile, charge, and gun. Four physical principles--governing the transformation of energy, the rate of change of projectile momentum, the linear buring rate, and the granulation geometry--suffice for a solution of the problem of computing projectile travel, speed, and chamber pressure as functions of time. These quantities may then be used for comparison with data obtained from experimental firings. Rockets. Rocket propellants may be either solid or liquid; if solid, the propellant should contain all the materials needed to maintain steady combustion; if liquid, the fuel and the oxidizer are kept in separate containers and come in contact only in the motor chamber. In either case, the process must be stringently controlled; a good pressure-time curve for a rocket is roughly trapezoidal, rising steadily to a plateau that is maintained during thrust and dropping rapidly at burnout. The mechanical strength of the chamber should not greatly exceed the maximum chamber pressure that can occur in practice, since the extra wall-mass can reduce acceleration. The shape of the nozzle or nozzles is a critical feature in the design of rockets. The contour chosen for the longitudinal section should steadily increase the speed of the gas into the throat, where it may flow at the speed of sound; beyond this point, the gas pressure declines and its speed increases. If gravity and air-drag are ignored, the motion of a rocket can be calculated by using an equation derived directly from Newton's second law of motion. This equation suggests that high-velocity rockets should permit exhaust gases to escape at high speeds and should carry a mass of propellant that is large in comparison to the mass of the rest of the rocket. A high initial propellant temperature will increase the chamber pressure in a rocket, which will in turn increase the thrust. However, too high propellant temperature will cause burnout to occur quickly and will have no substantial effect on the final speed of the rocket. Exterior Ballistics Exterior ballistics, in the classical sense, deals with the flight of projectiles moving under the influence of gravitational and aerodynamic forces. The study of certain types of projectile, however, may require that forces not treated in the classical theory of the subject be considered. Guided missiles, for example, may be acted upon by corrective forces, such as motor thrust or aerodynamic lift due to fin movement, when their trajectories depart from prescribed paths in space. Trajectory in a Vacuum. Galileo found that the path of a cannonball rolled off the end of a plank was the descending branch of a parabola, similar to the trajectory of a very heavy modern bomb dropped from an aircraft in horizontal flight. The trajectory of a projectile fired in a vacuum from an inclined gun barrel would include both the ascending and the descending branches of a parabola. Galileo assumed--incorrectly, because he was unaware of a small angle called the jump--that the elevation of the initial tangent to the trajectory was equal to the quadrant elevation of the bore. From his range-elevation relation he found, as Tartaglia had, that the maximum range was obtained when the elevation was 45 deg. He also found "that of other shots, those which exceed or fall short of forty-five degrees by equal amounts, have equal ranges." Drag. Galileo and Newton were both greatly interested in the force called air resistance, now usually called aerodynamic drag, which reduces the speed of a projectile. Galileo had compared the times of fall of oak and lead balls dropped from heights of "150 or 200 cubits" and found small but definite differences in their times of descent. He concluded that the deceleration due to drag varied with the projectile's shape, increased with its speed, and decreased with its density--more accurately, this should have been the modern sectional density, or mass divided by diameter squared. By dropping pellets from different altitudes, Newton found that the drag was apparently proportional to the square of the velocity of the projectile. He was aware that what is now called the drag coefficient depended upon the projectile's shape, but he regarded it as constant with respect to other variables. The first determinations of the drag of projectiles in flight were made by Benjamin Robins, using the ballistic pendulum. He found that the average value of drag-induced deceleration was about 80 times the acceleration of gravity. This demonstration amply established the importance of drag as a force affecting the motion of projectiles. Robins next undertook to determine drag as a function of speed. He obtained data approximately in accord with the Newtonian square law up to speeds of about 275 m/sec (900 ft/sec), but beyond this point he found great differences between his experimental results and those predicted by Newton's law. Robins, who made measurements up to speeds of about 520 m/sec (1,700 ft/sec), did not fully understand the significance of the differences between his measurements and Newton's predictions. It is now known that these differences chiefly result from the augmentation of the drag caused by the bow wave that forms at the head of a projectile moving at a speed near that of sound because of the compressibility of the air. In modern terms, the effects of the compressibility of the air are expressed through dependence of the dimensionless aerodynamic coefficients upon the \TMach number\t, which is the ratio of the speed of the projectile to the speed of sound. The drag of a projectile moving head on is now usually divided into three parts: bow resistance, due to air pressure at the head of the projectile; skin friction, caused by the friction of air moving along the middle portion of the body; and base drag, due to the under-pressure and disturbance of the air behind the base. At speeds slightly greater than that of sound, head drag can be diminished by a sharp, extended point; skin friction by a smooth, somewhat streamlined body; and base drag by a boattail. At Mach numbers of about 3, projectiles with slender, sharply pointed heads and boattails have about half the drag of more obtusely coned, square-based projectiles of the same diameter. The equations of motion of a particle acted upon by aerodynamic drag and terrestrial gravity can be written by using Newton's second law of motion. In the early 18th century, Johann Bernoulli of Switzerland examined the problem of a particle moving under the influence of gravity and drag proportional to the nth power of the velocity. He changed the equations of motion by substituting, for time, the angle of inclination of the tangent to the trajectory as the independent variable. The Bernoulli solution, obtained in 1719, was widely employed to compute trajectories during the 19th century and continues to have some application for the motion of projectiles fired at low velocities. Leonhard \TEuler\t of Switzerland was the first major writer on ballistics whose work was presented in analytical rather than geometrical form. In the mid-18th century he wrote equations of motion for a particle projectile and devised approximate methods of solving them that have been used repeatedly by later writers. A somewhat more convenient method was devised by James \TGregory\t about the same time and served as the basis for the important method for computing trajectories developed by American astronomer F. R. Moulton during World War I. At sufficiently great altitudes and ranges, exterior ballistics merges with the field of \Tcelestial mechanics\t. Problems encountered under such conditions fall within the province of a new science, sometimes called geoballistics. Projectile Stability. Most projectiles, such as bombs, artillery shells, and rockets, are somewhat elongated and have warheads placed toward the forward end of the body. Satisfactory fuse action for such missiles requires that they travel roughly head-on in their trajectories. This type of nearly head-on motion is called stable flight and is roughly synonymous with reasonably close trailing of some longitudinal line in the projectile along the tangent to its space path. Artillerists generally compare stability of motion with two other conditions, called instability and superstability; an unstable projectile may yaw violently or tumble, while a superstable projectile may maintain a fixed attitude in space regardless of how the tangent to its trajectory turns. Both instability and superstability are undesirable conditions for most conventional projectiles: for example, an unstable bomb may tumble and strike far short of its target because of excessive drag due to yaw; a superstable howitzer shell fired at a high angle of elevation may land base down, so that its point fuse may fail to operate. Stability in flight is ordinarily achieved by placing fins on the rear of the projectile or by giving it a rapid spin about a long axis by rifling in the bore of a gun. Bombs and darts are said to be fin-stabilized; rifle bullets and artillery shells are spin-stabilized. Both methods of stabilization are based on ancient principles: for example, arrows are finned with trimmed feathers, and flywheels resist forces that tend to cant their axes. Terminal Ballistics Terminal ballistics deals with the destructive actions and effects that occur at the end of the projectile's flight as an integral and undeformed body. The flight may end in one of two ways: the projectile may strike a solid obstruction, or its metal case may be broken by the explosion of a bursting charge. The phenomena of impact of a solid missile on a solid target develop as a continuous physical process; they have been extensively studied by flash radiography and are more readily predictable than those of the bursting of a high-explosive charge, whatever the nature of the surrounding medium. Impact Studies. Impact studies consider the mechanical impulse delivered by a solid projectile striking a target and the resulting internal forces, motions, and deformations affecting the two bodies. Traditional types of solid projectiles designed to achieve high penetration include steel-jacketed rifle bullets and armor-piercing bombs and shot. The projectile may strike into a target to some depth, leaving an indentation, or crater, or it may create an opening all the way through the target; a crater is said to result from a partial penetration, while a face-to-face opening is called a perforation. The mechanical impulse delivered to the target is greatest when the projectile's long axis is aligned on its path and the path is perpendicular to the face of the target. Empirical formulas for spherical projectiles indicate that the depth of penetration is proportional to the cube root of the square of cos A, where A denotes the angle between the path of the projectile and the perpendicular to the face of the target, but such formulas do not hold for impact angles close to the critical values at which a ricochet may occur. Ricochets occur more frequently as the impact angle increases; for firings at a given speed, the angle at which one-half of the rounds ricochet is called the ricochet angle. Experimental studies also indicate that the depth of penetration increases with the projectile's speed and its sectional density, the ratio of the projectile's mass to the square of its diameter; heavy, needle-shaped projectiles penetrate better than light, blunt ones. Detonation and Fragmentation. The physical action of high explosives is called detonation. Unlike propellants--in which the chemical reaction proceeds relatively slowly--the chemical reaction of high explosives can keep pace with the physical disturbance resulting from the reaction. The resulting narrow-reaction zone is called a detonation wave and can move in explosive materials at speeds as great as 6.3 km/sec (4 mi/sec). The gaseous products behind the front may have pressures of 50,000 atmospheres and temperatures of 3,000 to 5,000 deg C. If the wave strikes a solid material, it will deliver a mechanical impulse whose principal destructive effect is fragmentation. It has recently been possible to control the masses, sizes, shapes, velocities, and directions of motion of fragments by varying the characteristics of the case, for example, by corrugation or surface shaping. F. V. \TReno\t Bibliography: Corner, J., Theory of Interior Ballistics of Guns (1950); Fuller, J. F. C., Armament and History (1945); Kinslow, Ray, ed., High-Velocity Impact Phenomena (1970); McShane, E. J., et al., Exterior Ballistics (1953); Sterne, T. E., Introduction to Celestial Mechanics (1960); Sutton, G. P., Rocket Propulsion Elements, 3d ed. (1963).