Ballistics is the branch of mechanical physics that studies the motion of a projectile, understood as an inert body subjected to the force of gravity and the viscous friction of the physical medium of propagation.

The projectile has an initial velocity because it has been imprinted a force of impulsive type and does not have any type of artificial propulsion (i.e. not due to natural forces) that persists over time. Therefore it proceeds only by inertia and by the influence of natural forces (gravity, friction, winds).

The motion of bodies subjected to a continuous force (excluding gravity, friction and other natural forces) is not the object of study of ballistics, but of kinematics and dynamics. An example of this is rocket-powered missiles, whose motion is the object of study of kinematics and dynamics as long as the engine is running and becomes the object of study of ballistics when the engine stops working.

Missiles and rockets, therefore, assume ballistic behavior from the moment the engines cease to function, provided they are in a gravitational field. Of particular interest are intercontinental ballistic missiles, which reach considerable heights and have a very curved trajectory; in the downward phase, the trajectory follows in fact a paraboloid curve similar to that of a projectile of the same characteristics.

A similar behavior have also the semi-orbital bombs, or fractional orbit, which follow a ballistic parabola starting from an orbital position. The purpose of these devices is to evade missile interception devices, valid for suborbital missiles (see anti-aircraft defense). With regard to projectiles, ballistics is divided into three areas: internal ballistics, external ballistics, and terminal ballistics. Internal ballistics studies the propulsive factors and behavior of the projectile until it exits the barrel, external ballistics studies the behavior from the exit of the barrel to the point of impact, and terminal ballistics studies the effects caused by the projectile on the target.

History of ballistics

The mathematical study of shooting and therefore the birth of ballistics as a science can be traced back to the work of the Brescian N. Tartaglia (1499-1557), later taken up and developed by countless researchers, including G. Galilei and I. Newton.

The trajectory was originally considered linear, consisting of two straight segments, until Tartaglia demonstrated the curvature. Galileo later established that the trajectory is a parabola, limiting himself to the theoretical study of the function. Resuming Galileo’s work, Newton demonstrated (1723) that, although the trajectory in vacuum is a parabola, in air is a paraboloid curve with the descending line shorter than the ascending line and tending to the vertical.

This is due to the resistance offered by the air, which he proved to be directly proportional to the square of the velocity of the projectile, the density of air and the square of the caliber. This theory proved to be correct for subsonic velocities, around 250 m/s, while at higher velocities there are perturbations of the trajectory due to compression and rarefaction waves.

Newton’s statement must therefore be modified, replacing the square of the velocity with a resistant function, dependent on the speed itself. For the empirical study of ballistics and for the verification of theories, it was necessary to determine with sufficient accuracy the velocity of projectiles. This was made possible by Benjamin Robins, author of Principles of Gunnery, who invented the ballistic pendulum (1742). Important contributions to ballistic studies are due to Italians, especially Cavalli, Siacci and Bianchi.

Ballistics branches

Ballistics is so important to the study of the motion of projectiles fired from firearms that the latter became the main stimulus that led to the development of this science. Ballistics has since been differentiated into various branches of study that retain the prefix “ballistics” as related to firearms and its projectiles, although sometimes the use of this prefix is improper from the etymological point of view and is not always related to purely physical studies.

Ballistics is now divided into five main branches:

  1. Internal ballistics: studies the behavior of the bullet while it is still inside the muzzle, subjected to the forces generated by the ignition of the launch charge.
  2. Intermediate ballistics: studies the motion of the projectile in the immediate vicinity of the muzzle. It consists in the study of intermediate phenomena between internal and external ballistics.
  3. External ballistics: studies the motion of the projectile after it has been fired or launched. This is the actual ballistics, according to its definition.
  4. Terminal ballistics: studies the reaction of a body that comes into contact with the bullet.
  5. Forensic ballistics: is the discipline that includes the expert activities pertaining to firearms related to criminal investigations involving actions performed with firearms themselves.

Internal ballistics

Factors inherent in internal ballistics are primarily the barrel core, bullet weight and jacket, propulsive charge, and characteristics of the chamber and eventual case. With the exception of mortars, rocket launchers, and shotguns with split ammunition, all barrels have a rifled core.

The purpose of the rifling, which always has a helical pattern, is to give the bullet a rotary motion on the longitudinal axis, thus obtaining a gyroscopic stabilization that mitigates the effects of drift. In the case of short firearms, especially pistols, the bore is exactly cylindrical and the rifling has a constant pitch; as the power of the firing charge increases, in order to avoid maximum pressure peaks at the moment of the bullet notching in the rifling and instead obtain a thrust that is as progressive as possible until it exits the muzzle, the bore of the barrel assumes a truncated cone profile and the rifling is carried out with a progressive progression. In this way, the bullet encounters a progressively increasing resistance and the pressure is maintained at high levels until it exits the barrel.

Barrels made in this way also give better shooting accuracy and excellent ballistic performance, even with steel jacketed bullets. In large caliber artillery pieces, to avoid the enormous resistance to notching in the rifling and the rapid wear of the rifling itself, pre-cut bullets are used with helical grooves corresponding to the reliefs of the rifling. In any case, artillery projectiles, even those of medium caliber, are equipped with bronze rings, called force rings, which are designed to cut into the rifling and avoid direct contact between the barrel and the steel jacket.

In light artillery pieces and portable weapons, the ballistic behavior is also influenced by the characteristics of the case, which is generally made of brass. In these cases, in fact, it is the case that ensures the gas tightness when fired, following the expansion of the collar against the walls of the chamber.

Therefore, the degree of elasticity of the collar of the case and the amount of crimping, i.e. the tightening of the collar around the bullet, are important, as they make it possible to graduate the resistance to disengagement and consequently the value of the initial pressure. One of the most important factors in internal ballistics is the quality and quantity of the powder constituting the launch charge. Depending on the chemical composition, degree of fineness and external grain coating, powders have a higher or lower burning rate (liveliness).

Vivacious powders are suitable for shotgun and short arms cartridges, medium vivacity powders for rifles and machine guns and slower powders for artillery. For best results a powder should burn completely and progressively up to the exit of the bullet from the muzzle without giving excessive initial pressures and disruptive phenomena.

Certainly internal ballistics as a scientific discipline was born after external ballistics: if, in fact, the latter refers exclusively to the principles of mechanics, because it deals with the behavior of a mass, that of the “projectile”, in the gravitational field (in the presence of non-simple phenomena such as the resistance of the medium and the gyroscopic effect), internal ballistics focuses on the study of the combustion of the launch charge, on the consequent pressure trend inside the “barrel” of the firearm and on the induced effects that very often, beyond the prejudice regarding the result of the shot, call into question the safety of the shooter. The existence of various national firearms test benches (the Italian one is particularly famous) demonstrates the need to subject firearms to particular tests, which concern their “internal” behavior and which require, once passed, a special marking.

More than Galileo and his laws, internal ballistics refers to chemistry and thermodynamics: for this reason it is certainly a recent discipline. It should be added that the phenomena mentioned take place in transient regimes of very short duration, which greatly complicate the possibility of measurements and surveys. However, the fundamental purpose of internal ballistics is the measurement or prediction of the pressure trend inside the barrel. This is particularly important in weapons where the repetition of the shot occurs as a result of the “automatism” triggered by the previous shot. In fact, if, due to this automatism, the bolt were to retract before the residual pressure in the barrel due to the shot had dropped to acceptable values, there would be risks for the shooter’s safety, who would be hit by the high temperature pressure “dart” that would be released from the breech. The elements that contribute to the internal ballistic behavior of a system are: the type of launch charge (gunpowder), its primer and quantity, the boundary conditions of humidity and atmospheric pressure, the inertia of the projectile, the way it engages the rifling, friction, etc.

In principle, it can be said that the greater the “difficulty” of the bullet to exit the barrel, the greater and more dangerous will be the pressure inside it. In this case, the so-called “progressive” gunpowders are used, characterized by longer combustion and more gradual pressure increases, as opposed to the so-called “lively” powders. The situation is similar to what happens in the combustion chamber of a reciprocating engine: the piston is the projectile, the mixture is the launch charge, the spark plug is the ignition. The monitoring of the phenomenon is constituted by the relative pressure diagram, otherwise known as “indicated” diagram, whose relief allows to judge about the technical goodness of what happens there.

Even in the case of a spark ignition engine the combustion of the mixture must take place gradually: hence the extreme impropriety of the term “internal combustion engine”. When the mixture bursts, the engine behaves badly, because it hits the head: the release of energy is so sudden and violent that the piston can not keep up to transform it into work: so it is dissipated in the form of pressure waves of high peak that damage the engine because they break the protective veil of the lubricant and create hot spots that lead to absolute anarchy of successive combustions.

Hence, modern gunpowders are different from explosives, even though they share the same origin: they must moderate the release of energy: if the energy were released in a disruptive way, the barrel would burst. Therefore, modern gunpowders represent a compromise between the explosive action of nitroglycerine and the retarding action of cellulose: hence the diffusion of the so-called double base powders, nitroglycerine and nitrocellulose, which allow to obtain this compromise.

External ballistics

External ballistics concerns the behavior of the bullet from the exit of the barrel to the point of impact. Even in the so-called “straight shot” weapons, the trajectory is never a straight line, but a paraboloid curve that intersects in two points the line passing through the axis of the barrel. The accuracy of the shot is inversely proportional to the area of the scattering rectangle, that is the surface that includes the impact points of the shots, and is exclusively determined by the ballistic performance of the weapon and the ammunition.

The closer the center of the scatter rectangle is to the target, the more accurate the shot. Adjusting the shot is done by elevation and, when possible, by varying the launch charge. The most important factors in external ballistics are bullet shape and weather conditions, especially wind. The shape of the projectile must be as aerodynamic as possible and such that its center of gravity is approximately coincident with the geometric center; generally a tapered shape and tapering at the tail is chosen.

Regarding the influence of the wind, the most important phenomenon is the Magnus effect, from the name of the German physicist H. Magnus (1802-1870), who discovered that, when an air current strikes laterally a projectile rotating on its axis, cavitational phenomena occur that generate a thrust acting orthogonally both to the air current and to the direction of the projectile.

In order to understand the subject of external ballistics and in a certain sense its scientific necessity, it is convenient to start from the elementary problem: the behavior of a mass “launched” (in greek “ballo” means to launch, whence ballistics) with a certain initial velocity, independently from the way this launch is obtained: with a catapult, with a crossbow or the deflagration of a charge in the barrel of a firearm.

We know that if there were no gravity and other forces, the mass would continue to travel straight and at constant speed the trajectory impressed by the launch, according to what is stated by the first principle of dynamics. If we neglect all other forces and we admit that there is only gravity, the problem can be approached very simply considering that the initial velocity is decomposed in two components, one of which is constant horizontal and the other is uniformly decelerated vertical due to gravity. At the highest point of the trajectory the vertical component is zero and the corresponding part of initial kinetic energy is all transformed into gravitational potential energy. The trajectory is a mass independent parabola.

Then we consider the air resistance. This is a force that, due to the irregularity of the projectile shape, does not pass exactly through its barycenter where the force of gravity is applied. The result is the birth of a torque that tends to rotate the bullet, with obvious inaccuracy of the shot. To avoid this problem, the bullet is given a rotary motion around its main axis by the rifling of the barrel. Due to the principle of the gyroscopic effect, the bullet no longer rotates in the vertical plane, but tends to drift, i.e. to deviate from the plane of the theoretical shooting parabola, with an error called drift error. It is possible to correct this error with aiming and sighting devices.

Terminal ballistics

Terminal ballistics studies the effects of the projectile on the target and the causes behind them, such as mechanical characteristics and residual velocity at impact. In the case of non-explosive projectiles, the main factors are the residual energy expressed in kilogrammeters, the penetration coefficient and the consistency of the bullet tip.

The study is often aimed at maximizing the stopping power of ammunition, with regard not only to the caliber, but above all to the structure of the bullet, and this is even more so when used for personal defense purposes. In fact, for personal defense it is not important that the bullet be lethal, but rather that it stop the aggressor. Terminal ballistics therefore studies the deformations that the bullet undergoes at the moment of impact, any fragmentations that must be avoided, and the shape that the bullet assumes depending on the type of ammunition (jacketed, armored, half jacketed, naked, etc.).

A bullet that penetrates the body by passing through it transmits only part of the kinetic energy it possesses to the target, and therefore has a low stopping power. In this case, the projectile can be lethal but often does not transmit to the target the shock necessary to stop it instantly and avoid a dangerous hostile response. The stopping power is in fact related to the amount of kinetic energy present at impact and the percentage of this that is transmitted to the target.

The study of the terminal ballistics of a projectile is therefore important: a projectile that deforms on impact, assuming the classic mushroom shape, will be easily stopped by the body of the target that will absorb all the kinetic energy; the shock effect and therefore the stopping power will be maximized. This is the case of hollow point bullets, which are forbidden by Italian law for personal defense purposes, while armored bullets are allowed, similar to those used for military purposes, which have the characteristic of being less lethal.

Depending on their intended purpose, bullets are divided into armor-piercing, expanding, and frangible; on living targets, the most devastating results are obtained with expansive, soft-tipped, possibly perforated bullets. Small caliber bullets animated by very high velocities, greater than 1000 m/s, give rise to explosive phenomena due to the pressure wave on impact against living tissue and are therefore much more destructive than bullets of higher caliber, but slower.

All fast projectiles produce an exit hole much larger than the entry hole, behaving similarly to expansive projectiles. Explosive projectiles exhibit impact behavior consequent to the type and quantity of the charge. In this regard, the most important phenomenon is the Munroe effect, universally exploited in hollow-charge armor-piercing projectiles.

In 1888 the American chemist C. E. Munroe (1849-1938) discovered that when the charge is placed in such a way as to present a conical cavity at the tip of the projectile, on impact the energy of the explosion is concentrated in a small central area and generates a dart capable of perforating ten times the thickness of steel.

Forensic ballistics

Forensic ballistics is a branch of forensic science that includes investigations aimed at the reconstruction of the events related to a crime in which a firearm was used, aimed at the definition of responsibility and the imposition of punishment. Forensic ballistics is based on the principle that all firearms have indelible differences due to the different mechanism with which they were manufactured.

In the cases in which a firearm has been used, with consequent lesions or death, a correct medico-legal diagnosis, even if based on the careful examination of the “basic” data (number of exploded shots, distance of shooting and reciprocal position between the wounded and the victim), can sometimes prove to be incomplete or insufficient in the absence of an integrated evaluation with the results of investigations commonly considered to be of a more exquisitely criminalistic nature, such as the examination of the weapon and its mechanics, the definition of the number of unexploded shots in the magazine, the identification of the caliber of the exploded bullets, as well as the interpretation of environmental and testimonial findings.

Specific to forensic ballistics are investigations for:

  • the identification and description of the place where the event occurred;
  • the examination of damage from ballistic agents in environments and on vehicles;
  • the search, collection, preservation and identification of findings of ballistic interest;
  • the examination of the weapon, the ascertainment of its characteristics and functionality;
  • identification of the shooter;
  • the evaluation of the shooting distance.

In addition to these, in proposing reconstructive hypotheses of the event, due account must be taken of the certainly not secondary and classic medico-legal themes of:

  • evaluation of the time of death and/or injury, of the cause and of the means used;
  • the survival time and the possibility of autonomously carrying out actions or movements after the injury;
  • type(s) of weapon(s) used, caliber, number of rounds fired, still firing distance and mutual position between victim and shooter.

The forensic ballistics suffers however of some difficulties of analysis with particular types of weapons whose bullets slide through plastic coatings that prevent the contact with the barrel. For these types of weapons there are other analyses that can reveal their sources.