Ballistics of Modern Firearms

The history of the firearm stretches back to the middle ages. An exact date is uncertain, but there are reported instances of firearms being used as early as the middle of the 14th century. At that time guns were little more than a wrought iron or bronze tube a few feet in length, closed at the breech and left open at the muzzle [1]. This “hand cannon” was loaded by pouring gun powder down the barrel packed tightly with a rod or stick, then inserting a round projectile on top. It was fired by igniting a small amount of powder placed in the touch hole-a hole drilled into the breech-that in turn ignited the main charge and propelled the projectile through the barrel. These early weapons likely served little practical purpose other than frightening whomever they were being fired at; they frequently failed, producing a great deal of smoke and noise with wild inaccuracy.

However unreliable, firearms proved themselves useful, gaining wider usage as new designs and technologies emerged (see Fig. 2.). The most important inventions that led to the creation of modern firearms were the percussion cap in 1807 and the cartridge in 1851 [2]. These two innovations, more than any others, paved the way for the current state of firearm technology.

The modern firearm exists in a myriad of forms ranging from simple single-shot rifles to complex self-loading handguns. In its most basic form, the firearm consists of a barrel, an action, a firing mechanism, and a stock or grip.

Modern guns fire an improved version of the cartridge-type ammunition invented in 1851. This type consists of a metal tube called the case or casing, a bullet, a primer and a smokeless powder charge. Put together, this type of cartridge is referred to as center-fire ammunition.

Most firearms operate by exploiting the same basic physical and chemical laws – it is therefore useful to study these principles independent of the form that the firearm takes. The study of these principles is called ballistics and consists of three parts: internal, external, and terminal.

Internal ballistics is the study of the flight of a projectile inside a firearm. This term is used by experts to refer to all aspects of the internal workings of a firearm from gases and pressure to rifling and bore erosion. As such, internal ballistics is a very large subject. Many books have been written to explain the complex physical and chemical processes that internal ballistics entails. Our main concerns when looking at internal ballistics will be the workings of the firing process, the purpose of rifling, and the causes of recoil.

The firing of a bullet by a rifle is the first step in the ballistics process. The powder charge contained within the cartridge ignites by means of a percussion explosive contained within the primer. A gun is fired by pulling the trigger which releases the firing pin from its cocked position allowing it to collide with the primer. The primer ignites and shoots a 2000 degree C stream of fire into the cartridge, igniting the main power charge. As the powder in the cartridge burns, it produces hot gas which causes pressure to build in the cartridge. As the pressure and temperature inside the cartridge increase, the rate of combustion increases, building up higher and higher pressure. At a certain point, the pressure in the cartridge exceeds the friction that holds the bullet in place, and it begins to move. The powder continues to burn, increasing the pressure in the chamber, until it reaches a maximum when the bullet is a short distance down the bore. As the bullet moves down the barrel, it accelerates, and the pressure decreases slightly as a result of the expanding volume behind the bullet [3].

When the bullet leaves the muzzle of the gun, the pressure has subsided significantly, but has not completely dissipated. The rush of hot gas expands dramatically when it leaves the barrel, causing a pressure wave to develop in the air. It is this pressure wave that we perceive as the loud bang produced when the gun is fired. Silencers change the rate of expansion of this gas, making the shot quieter. This principle can also be found when considering balloons. If you pop a balloon it makes a loud bang as the air expands, but if you were to untie the same balloon and let the air out slowly, considerably less noise is heard.

Interestingly, all of the powder does not burn in the cartridge. The explosion caused by the extreme pressure expels some powder out of the muzzle with the bullet; this burning powder produces muzzle flash. The resulting flash is consequential, because it can cause a shooter to temporarily loose vision capability when firing a gun at night. To remedy this problem, manufacturers of smokeless powder created a faster burning powder; thus less unburned powder is ejected from the barrel, leading to less muzzle flash [2].

When a football player throws a ball, he tries to spin it on an axis parallel to the direction of motion. This action causes the ball to stabilize through gyroscopic forcing, and keeps it from tumbling end over end. A similar process happens when a bullet is fired from a gun. The bore of a gun has small grooves cut on the inside, termed rifling; hence the name rifle. The rifling spirals down the inside of the bore and imparts spin on the bullet as it travels down the barrel (see Fig. 1). This spin contributes a great deal to the accuracy of modern firearms [4].

By measuring the distance between the lands of the rifling inside the bore, one can determine the caliber of a firearm. This measurement, which is usually less than one inch, is commonly measured in hundredths of a inch. For example, a 30 caliber bullet would correspond to .300 inches. Caliber can also be measured in millimeters. For instance, a 9mm handgun has a 9 millimeter bore diameter. It is important to note that barrels are made for a specific caliber, but also chambered for a specific cartridge. If, for example, you had a rifle that fired .300 Remington Ultra Magnum cartridges, you would not be able to use it with a .300 Winchester Magnum, even though it is the same caliber as the Remington.

When you fire a gun, the bullet leaves the barrel at a high velocity, on the order of thousands of feet per second. The laws of physics say that this bullet has a kinetic energy that is determined using the formula:

where K= kinetic energy, m= mass of the bullet, and v= velocity of the bullet.

According​ to this formula, energy is directly proportional to bullet size, and the square of velocity. Though especially significant when considering external and terminal ballistics, this relationship is also important when considering recoil. Recoil can be thought of as the reaction caused by the creation of a bullet’s kinetic energy. Newton’s third law of motion states that for every action there is an equal and opposite reaction. Therefore, when a bullet is given a forward energy, there must be an equal energy in the opposite direction; the effect of this reverse energy is what is called recoil.

A number of factors influence recoil, the most obvious of which are the caliber and muzzle velocity of a bullet. These two quantities determine the magnitude of the recoil, but not the qualitative effect of recoil. To determine how the recoil feels, you must take into account variables such as acceleration time, firearm mass, and shooter mass. The acceleration time is proportional to the period over which the shooter must absorb the recoil: the shorter the time, the harder the “kick” felt by the shooter. Recalling the components of kinetic energy, we find that a heavier firearm results in less recoil absorbed by the shooter; likewise, a heavier shooter will be less likely to be knocked over by the recoil.

Many of the terminal characteristics of bullets are improved by increasing bullet mass and velocity; however, too much recoil leads to shooter discomfort. As a result, a great amount of engineering mitigates the effects of the recoil.

External ballistics studies the behavior of bullets from their departure from the muzzle of a firearm until the striking of a target. The projectile motion problems found in any introductory physics class are examples of external ballistics. The most basic aspects of these problems involve the mass of the projectile, its velocity, and the angle of departure. Using these figures, one can calculate the time of flight, trajectory, and range, albeit only in textbook fashion. In the real world, factors such as air resistance, sectional density, and bullet shape substantially affect external ballistics.

The most significant factor affecting projectile motion aside from gravity is air resistance, which is the result of air molecules colliding with the leading edge of a projectile. The air molecules strike the projectile, exerting an opposing force which slows the projectile down. This decrease in velocity causes the projectile’s trajectory to droop more abruptly, resulting in a shorter maximum range. This effect is termed “drop”.

In air, the speed of sound is approximately 1100 feet per second. Typical bullet velocities for rifles, on the other hand, are on the order of thousands of feet per second. This means that rifle bullets travel faster than the speed of sound and will cause a sonic tearing as they travel through the air. This has important complications for silencers because it means that to get a firearm to be sufficiently quiet, you must have bullets that travel at a velocity less than the speed of sound. As a result, silenced rifles are largely ineffective at long range because they experience more drop.

One of the most important factors is determining the degree to which air resistance will affect a projectile is cross-sectional density. One can think of cross-sectional density as a measure of the carrying power of a projectile. For example, if you throw a ping-pong ball through the air, it will experience some air resistance proportional to the cross-sectional area of the ball that comes in contact with the air. The ping-pong ball, which has a small mass, is likely to be affected a great deal by the air resistance it experiences as it travels. An equally sized lead ball thrown through the air experiences an effectively non-existent air resistance, due to the ratio of the lead ball’s mass to its cross-sectional area, which is much greater than the ping-pong ball’s. It is very important when considering the performance of a projectile to keep in mind the properties of cross-sectional density [2].

To put a quantitative value on the degree that air resistance affects a certain type of bullet, engineers developed the ballistic coefficient:

where C= ballistic coefficient, W= weight of bullet, i= form factor, and d= diameter of the bullet.

Simply put, the form factor corresponds to the shape of the bullet and is inversely proportional to the degree of aerodynamic characteristic of the bullet, i.e. a pointy bullet has a higher number and a flat one has a lower number [2].

By revealing the degree to which a bullet will retain its velocity, the ballistic coefficient can indicate the amount of drop of the bullet, and therefore its resulting accuracy. Although it is possible to design bullets with very high ballistic coefficients, this can be both beneficial and detrimental to bullet performance in terminal ballistics.

When studying terminal ballistics, engineers and scientists are interested in determining the effects of a projectile on a target. The important factors to consider are energy, penetration, and expansion.

Ironically, the same energy that we are trying to minimize in the recoil found in internal ballistics should be maximized in terminal ballistics. The idea is for the bullet to hit the target with as much energy as possible. Recall the equation for kinetic energy from Equation 1. This indicates that the higher a projectile’s velocity upon hitting a target, the more energy is carried into the target, thus causing more damage.

In the real world, energy is conserved, whereas in the movies it can be created out of thin air. Recall that the kinetic energy released when a bullet hits a target is less than the kinetic energy that the gun imparts on the bullet when it is fired – this is a necessary consequence of air resistance and the associated decrease in velocity. That means that if a bullet were to knock a bad guy off the ground and through a window, it would probably break the shooter’s shoulder while sending him flying backwards.

Equally important to the energy carried by a bullet is the amount of penetration that can be achieved when it hits a target. African big game hunters will tell you that it is vitally important for bullet to be able to penetrate a target because energy alone cannot take down a large animal. After all, an elephant can absorb much more energy as a result of being hit with a bullet than a shooter can through recoil. A bullet can do no damage to a target if it cannot penetrate the armor.

To remedy this, engineers developed a couple good solutions, the most obvious of which simply makes the bullet sharper. Indeed, this is a very good solution because it also makes the ballistic coefficient higher, resulting in all the benefits of increased velocity. Another option is to make bullets harder. A hard bullet will penetrate without breaking into small pieces that are more easily slowed down.

Unfortunately, bullets that penetrate too well also tend to pass through objects without causing much damage. A gun that just makes neat little holes is only useful for target practice; what is needed is a controlled way for a bullet to penetrate an object and then release all of its energy inside.

The solution to this dilemma is expansion – we want our highly aerodynamic bullet to enter the target and deform. The deformation will cause the ballistic coefficient of the bullet to decrease, thus transferring the energy of the bullet to the target. In an ideal situation, the bullet penetrates the target to a desired distance and stops. Engineers have created a plethora of different types of bullets with different terminal characteristics for different applications.

Most types of bullets are variations of the jacketed bullet, meaning that they have a soft dense core surrounded by a hard shell. As usual, there are many exceptions to this statement. Small caliber rifles and handguns frequently shoot bullets constructed entirely of lead, and shotguns use a completely different ammunition design altogether. Nevertheless, here we will concentrate on the most common jacketed variations.

The majority of the mass of a jacketed bullet comes from the core. Lead is frequently chosen for the construction of the core because it is dense (resulting in a greater sectional density) and malleable, meaning that it deforms well without breaking. However, lead bullets also have negative characteristics. The malleable nature of lead causes it to slip on the rifling resulting in spin rates that are not uniform across firings, reducing consistency of performance and accuracy. In addition, lead does not maintain shape well during the internal and external ballistic phases, and lead deposits can coat the mechanical parts of a firearm, resulting in jams and misfires. To combat these problems, jacket bullets have been designed with a shell made of a more rigid material, such as copper, that allows the rifling to have a stronger grip, maintaining the shape of the bullet during flight without leaving harmful deposits.

Each type of jacketed rifle bullet has different terminal characteristics based upon the shape of the core and jacket. For example, in full metal jacket bullets, or FMJ’s, the core is completely surrounded by the jacket – the result is a bullet that experiences little or no expansion upon impact. These bullets are used in situations where penetration us the primary objective, such as shooting through armor plating. Soft point (or round nose) bullets have a jacket that is open at the tip, allowing the core to be exposed. This type of bullet expands upon impact and is frequently used in situations where a medium rate of expansion and little fragmentation is desired, such as hunting larger game. Hollow point bullets, on the other hand, have a high expansion rate and a low penetration depth due to a hole that is drilled into the end of the bullet. These bullets are used in situations where damage and fragmentation are the primary concerns, such as when hunting small game.

These bullets are only a small fraction of the bullets in use. There are many other types used by the military, police, and private citizens for combat, self-defense, sport shooting, and hunting purposes. Interestingly, the wide variety of bullets gives an indication of the many benign and beneficial uses of firearms. Though it may be easy to think of guns as mere killing machines, firearms are really used for many other purposes besides causing harm.

Guns are nothing more than sophisticated, specially engineered machines. Through the study of external ballistics, engineers can develop long range bullets which strike the target with great precision. Similarly, external ballistics allows engineers to create more aerodynamic firearm projectiles. Finally, the science of terminal ballistics helps engineers contrive bullets capable of striking the target with a variety of desired effects. This great diversity of bullet types allows the firearm to be used in a multitude of applications, ranging from sport hunting to military combat. In the final analysis, it is important to remember that firearms possess no more disposition for harm than an automobile or an airplane. Our fear of guns comes not as a result of the science and engineering that produces them, but as a result of the psychology of those that use them for malicious purposes. Indeed, firearms are used for many benign and even beneficial purposes. In the right hands they can easily protect life rather than take it away.