I  INTRODUCTION
It is certainly necessary to go back a long way in time to find the first motivations of the man who pushed him to send an object of any kind, but preferably blunt or sharp, on a fellow human being (act of war) or on an animal (act of hunting) with the aim of defense or aggression.
The main motivations were probably the impossibility of approaching the target (a fierce animal, for example) or the desire to stay at a distance from the target (a dangerous animal or adversary)
The entire history of ballistics can be summed up in one objective : to send a bullet as far and as accurately as possible at a target.
In this overview of ballistics, we will limit ourselves to talking about firearms, which are far from being the only way to send a bullet, since there are many devices that can serve this purpose.
Indeed, any system capable of transmitting its potential energy in the form of kinetic energy to a bullet can satisfy the need to reach a target placed at a certain distance. One only has to think of bows, crossbows, launchers using a spring, compressed air or even electrical energy which, in this last case, allows very high speeds to be reached.
In this presentation, we will limit ourselves to firearms and consider their ballistics classically broken down into three main parts:
 The internal ballistics of weapons that deals with the phenomena that occur in the chamber and barrel of the weapon until the exit of the bullet.
 The external ballistics which studies the aerial trajectory of the bullet.
 Terminal ballistics which talks about the interaction between the bullet and the target. When the target is a living organism, or a dead one for that matter, we talk about lesion ballistics.
Very often, the external ballistics is itself split into two parts:
 Intermediate ballistics, which is interested in what happens between the muzzle of the weapon and a few tens of centimeters further on, at the moment when the bullet leaves the barrel. We will see that the universe in which our bullet evolves, fortunately briefly, is tumultuous and chaotic due to the action of the gases that expand and cause some damage.
 The external ballistics which begins at the end of the intermediate ballistics, when the gases no longer have any influence on the bullet, and ends when the bullet reaches the target, where the terminal ballistics begins.
Let's start at the beginning and talk a bit about the internal ballistics of weapons.
II  INTERNAL BALLISTICS OF WEAPONS
This is a complex field. Although the equations governing the behavior of gases inside the gun may seem perfectly abstruse to the uninitiated, it is however possible to understand, even if only qualitatively, this part of ballistics and to leave to the real specialists the pleasure of integrating their differential equations.
II1  BASIC CONCEPT
In firearms, the energy produced by the deflagration of an explosive substance, commonly called "powder", is used to propel a bullet. This substance, by definition, is capable of releasing its potential energy in a very short time, during a chemical reaction, in the form of a large quantity of gas at very high temperature. It is these gases that will propel the bullet out of the gun.
Before going any further, let's see, schematically, how a firearm works.
• A firearm is primarily a tube open at one end (usually)
Whatever its type and its external shape, a firearm is composed of a barrel which is nothing but a tube open at one end to let the bullet out, obviously. The opposite end, called "chamber" is where the explosion of the powder occurs. The back of the chamber is closed by a metal piece called a breech. On modern small arms, the sealing of this part of the barrel is mainly ensured by the ammunition case.
SIMPLIFIED PRINCIPLE OF A FIREARME 

Figure II1/1 
• Cartridge
The evolution of techniques made it possible to gather the propellant charge, its firing system and the bullet in only one component : cartridge of which one will find a diagram below.
DIAGRAM OF A CARTRIDGE 

Figure II1/2 
Below are some cartridge models and details of a bullet for modern weapons.
HANDGUNS CARTRIDGES SAMPLES 

Figure II1/3 
CARTRIDGES FOR LONG GUNS 

Figure II1/4 
SOME TYPES OF BULLETS 

Figure II1/5 
The bullets have various shapes and compositions depending on the nature of the target and the type of bullet/target interaction required, while respecting the laws of aerodynamics.
At the moment of the explosion of the powder charge, the gas pressure is applied on all the walls of the chamber and on the back of the bullet, the base, which, with a diameter practically identical to that of the interior of the gun, ensures a gastightness while presenting less resistance than the walls of the chamber. The bullet is thus pushed by the gases towards the mouth of the barrel according to the piston principle.
One will find, below, a diagram representing a cut at the level of the breech, chamber and beginning of the barrel with a cartridge at the time of the departure of the blow. The bullet has already taken the rifling.
LONGITUDINAL SECTION AT THE LEVEL OF THE BREECH 
1) Firing pin  2) primer  3) flame jet from primer  4) ignited powder  5 case  6) bullet

From : "Munition für Leoichtwaffen, Mörser
und Artillerie". Ian V. Hogg.
Motor Buch Verlag. 
Figure II1/6 
• Weaponmunition matching
In the weapon/ammunition couple, the launcher (the weapon) is developed around the ammunition according to the following algorithm:
ADAPTATION OF THE GUN TO THE AMMUNITION 

• Smooth barrel, rifled barrel
The internal part of the barrels, called bore, is generally presented in two forms: smooth or helically rifled, with constant or variable pitch. These rifling allow to apply to the bullet, during its course in the barrel, a moment of rotation with the aim of suppressing its erratic behavior during its course in the air.
SOFT BORE VS RIFLED BORE BARRELS


• Shape of the bullets has changed with time... and the need of greater precision
In the early days of firearms, bullets were made of lead and were spherical in shape. A full and homogeneous sphere has a center which is, at the same time, geometric, of gravity and of symmetry.
The principle was simple. The inside of the barrels was smooth. The caliber of the cannon was not given according to its diameter, but rather, and this until about the 19th century, by the number of lead spheres that could be cast for the diameter of the cannon in an old pound of lead (489.5 g). This old habit is still found today for smoothbore shotguns. For example, the 12 gauge has a theoretical diameter of 18,5 mm and one could cast 12 spheres of lead for this gauge which is larger than the 16 gauge.
• The reason for the grooves
The first helical grooves are mentioned in 1476. Straight grooves are mentioned in 1498.
The ballistic experts of the time quickly realized the poor performance of smoothbore guns and their spherical lead bullets. The range of these weapons was poor and their accuracy decreased rapidly with distance. New solutions were needed.
During their journey in the smooth barrel, the spherical bullet, because of their design, were subjected to dissymmetrical friction forces that varied from one shot to another. They came out of the barrel with a rotational movement on themselves which, by interaction with the air, made their trajectory leave the shooting plane. They were endowed with an "effect" similar to that observed in ball games such as golf, tennis, soccer or other. The difference being that, in the field of sport, this effect is sought and controlled whereas in the old weapons it was random because the friction due to the interaction between the ball and the core of the barrel was not identical from one shot to the next. The idea was therefore, since there had to be an "effect", to foresee it by imposing a rotation, or not, on the bullet thanks to helical or rectilinear rifling. In the case of helicoidal stripes, the trajectory always went out of the trajectory but one knew from now on in which way and in which proportion. It was then enough to provide the sighting system with an abacus taking into account this deviation according to the range.
The idea of creating balls more aerodynamic than a sphere and with a more predictable trajectory led to the realization of bullets with elongated, oblong shapes and whose front has a more or less pointed profile.
One of the particularities of these bullets is that they do not have a center of symmetry but an axis of symmetry according to their length. When fired from a smoothbore barrel, they are perfectly unstable. They tilt or spin during their aerial trajectories.
Among the various methods of stabilizing a bullet, one chose to use the gyroscopic effect which transforms into a precessional movement the tendency of this bullet to tip over as soon as it leaves the barrel. For this purpose, it was necessary to make the bullet turn at high speed around its longitudinal axis. One thus traced helicoidal stripes in the core of the barrel in order to give it a rotational movement that it will keep throughout its trajectory. We will have the opportunity to evoke this stabilization by gyroscopic effect in the chapter "external ballistics".
Note : not all weapons that fire spinstabilized bullets have a rifled barrel. Some of these weapons have a barrel with a polygonal rather than cylindrical bore.
• How a weapon works
When we talk about the functioning of a weapon, we are of course talking about launching a bullet but also, in the case of most modern automatic or semiautomatic weapons, ensuring its feeding.
The means of propulsion
The means that can provide a weapon with the energy necessary for its operation are varied. We can, for example, quote some of them:
 Compressed gas: air, nitrogen, helium or hydrogen depending on the desired speed. In addition to the air compressed weapons
, it is also a means of propulsion used in test gun of laboratories ;
 Compressed springs, rubber cords under tension ;
 Electricity. We use the force generated by an electromagnetic field ;
 Explosive substances.
It is the propulsion with explosive substances that is mostly used, hence the name firearms. If it is not the simplest, it has become the most practical and allows the operation of both highpowered weapons and others of low volume and easily portable.
The pressure of the gases generated by the combustion of the powder being the principal means used for the operation of firearms, we will stop one moment on this, or rather, on these powders.
II
 2  GUN POWDERS
Powders are explosive substances whose combustion reaction, i.e. the release of their energy, is sufficiently low (compared to other types of explosives) to be used for propulsion purposes. These substances are transformed according to the deflagration regime. Their linear transformation speed is of the order of a few hundred metres per second.
This explosive behavior is generally due to combustion, i.e. a redox reaction. The particularity is that, in this reaction, the oxygen necessary for the combustion is not mainly borrowed from the ambient air (the reaction would be too slow) but is integrated inside the powder, or even the active molecule. This particularity explains the speed of the reaction.
• The black powder

The Arab author Ab Allah was the first to mention saltpetre (13th century). Everything suggests that the Chinese invented it in ancient times and used it for fireworks.
Almost at the same time, Roger BACON, would have given a formula of the black powder.
The history leaves us with the memory of the German monk, Berthold SCHWARZ, who would have lost his life while trying to use it as a means of propulsion in the bombardes. A legend claims that the Devil still laughs about it.
Black powder is a mixture of potassium nitrate or sometimes sodium, sulfur and charcoal. The proportions of these various elements, ground very finely, being variable according to the desired vivacity, one should rather speak about "black powders". 
History seems to have remembered the sad date, for France, of the battle of Crécy (1346) with regard to the appearance of black powder on the battlefield.
The English must have had two or three cannons that were noticed by the troops and the observers more for their noise than for their effectiveness, because, as the latter noted, once they had recovered from their emotions: "The Lord and the gentle Virgin Mary be praised, they did not hurt man, woman or child.
It was used for nearly five centuries, until Mr. VIEILLE invented smokeless powders (nitrocellulose powders).
Black powder is no longer used today except by sport shooters with antique weapons and by some manufacturers of rubber bullet cartridges. It should be kept in mind that black powder is relatively delicate to use. Indeed, it is an explosive substance whose transformation speed is of the order of 900 m/s, i.e. close to the limit that classically separates progressive explosives operating in the deflagration mode, of which powders are a part, and high explosives which transform themselves in the detonation mode. Depending on its confinement, black powder can pass quite easily from the progressive mode (deflagration) to the breaking mode (detonation) with harmful consequences on the material and, possibly, on the shooter and the close environment.
• Modern powders (called smokeless)
They are often referred to as "smokeless powders" although they are not powders and are not really smokeless.
In these propellants, the explosive properties of nitrocellulose, a substance obtained by the action of nitric acid on cotton, are used.
By nature, nitrocellulose is an explosive, and therefore unusable as is. To ensure slow combustion, control the speed and ensure good stability, nitrocellulose is gelatinized with an etheralcohol mixture.
• Single and multibase powders
There are two main families of smokeless powders: the "simple bases" and the "multibases".
In single base powders, the only active product is nitrocellulose. All other products in the composition are incorporated only to ensure stability and control of the burning rate.
In the family of multibase powders, we distinguish between "double base" and "triple base" powders. In these two types of multibase powders, nitroglycerin is added to increase the energy level of the powder.
Doublebase" powders produce less smoke and have a better energy yield. However, they have the disadvantage of eroding the barrels more quickly because of their very high combustion temperature.
To overcome this disadvantage and to find the qualities of "single base" powders, while keeping the easy manufacture of "double base" powders, a third base, nitroguanidine, is added. We obtain a combustion temperature similar to that of the "single base" while keeping a great propulsive potential.

• The "powders" are in the form of grains
Contrary to their name, powders are in the form of grains of various shapes. Combustion, and therefore the release of gases, takes place on the surface of the grains. It is easy to understand that the larger the surface of the grain, the greater the quantity of gas released in a given time. The shape of the grains of the powder, in addition to its chemical composition, greatly influences its vivacity, i.e. its speed of combustion.
• Influence of the shape of the grains
For a given amount of material, the geometric shape with the minimum surface area is the sphere. Starting from this lower limit, the powder grains are given geometric shapes with increasingly large surfaces (parallelepipeds, flakes, sheets, full cylinders, cylinders with a central hole or multiple holes along the longitudinal axis).
Its different shapes allow to have powder grains whose surface varies or remains practically constant during the combustion phase.
GUN POWDERS  TYPES OF GRAINS 


Miltirary Ballistics  GM Moss, DW Leemings, CL Farrar

• Mechanical characteristics of the powder grains
The powder grains must have good mechanical resistance because, during their combustion and pressure rise, they will be subjected to significant mechanical stress. They must not break, otherwise their shape will obviously be modified as well as their combustion speed and, ultimately, the vivacity of the powder.
• Powder ignition
The combustion of the powder is initiated by an ignition system. For smallcalibre weapons, a primer placed in the base of the case is used. This primer contains a primary explosive which is by nature not very powerful, compared to other explosives, but which is very sensitive to shock and, in general, to external constraints.
Among the explosives used were lead styphnate, otherwise known as lead trinitroresorcinate, and mercury fulminate, which was relatively unstable and fell into disuse.
• Ignition mechanism
When the primer is struck by the firing pin, the deflagration of the explosive it contains projects hightemperature gases and incandescent particles that will ignite the powder. Hot gases will be emitted on the surface of the grains.
Ideally, all the grains of powder should be ignited simultaneously, the pressure in the chamber should rise rapidly to its maximum, remain there until the powder is completely burnt and then relax from that moment. Thus the speed of the bullet would be optimized.
In reality, this is not the case. A "smokeless" powder gives a pressure that increases progressively with time and remains at this maximum for a very short time.The combustion of the powder is never complete.
Moreover, there are pressure waves, originating in the regions where the powder ignites, which propagate to the limits of the case and the base of the bullet where they are reflected and can interfere at certain points. The pressure field in the chamber is not homogeneous and this very complex phenomenon can influence the mechanical stresses on the walls of the chamber or case and the burning rate of the powder grains.
In any case, after ignition, the pressure in the chamber increases and the hot gases push on the base of the bullet. The latter will not move as long as the force generated by the gas pressure on its base is less than that due to the static friction of its crimping on the case, which is an important factor in the initial rise in chamber pressure.
II3  THE MOTION OF THE PROJECTILE IN THE BARREL
From the ignition of the primer, it is interesting to study the temporal evolution of three phenomena in the chamber and barrel of a weapon.
As the pressure in the chamber continues to increase, the bullet begins to move forward and is faced with new friction that will present a resistance to its movement.
This resistance depends on the type of stabilization chosen for the bullet. For bullets stabilized by rotation, the most important resistance to advancement in the ballistic cycle will be the taking of scratches.
Indeed, in order to communicate to the bullet the rotation torque allowing it to reach its optimal rotation speed, the rifling of the barrel's core must, for smallcalibre weapons, penetrate relatively deeply into its jacket. The shells, on the other hand, are equipped with a belt of ductile material, one of whose roles is to interact with the rifling of the barrel.
RELATIONSHIP BETWEEN PRESSURE AND BULLET POSITION 

Military Ballistics  GM Moss, DW Leemines, CL Farrar 
Figure II3/1 
The movement of the bullet results in an increase in the volume of the combustion chamber, a phenomenon that should lead to a decrease in pressure.
But the speed of combustion of the powder increases rapidly with the pressure and very high values are reached before the movement of the bullet. The result of these two antagonistic phenomena is the rapid attainment of the maximum pressure in the chamber. This maximum usually occurs shortly after the rifling is taken, whereas ideally it should occur at the moment of rifling.
CORRELATION BETWEEN PRESSURE, BULLET VELOCITY AND POSITION 

Figure II3/2 
• Friction against the walls of the barrel
Since the bullet must be sealed from the tube core, friction is important. In rifled barrels, there is also friction between the bullet and the stripes. We generally consider three types of friction:
• Force required to take grooves ;
• Frictional force, metal/metal, due to the movement of the bullet in the barrel ;
• Frictional force, metal/metal, due to the movement of the bullet in the barrel.
In the diagram below, the author breaks down the friction into its different components. For our part, in the following, for the sake of simplification, we will call FR the resultant force of all the friction which opposes the advance of the bullet. It is represented by a vector which is collinear to the force due to the gas but in the opposite direction
FRICTIONAL FORCES IN A RIFLED BARREL 

Interior Ballistics of Guns  M. Krier, M. Summerfield

Figure II3/1 
The frictional forces are at their highest when the grooves are taken.
FRICTIONAL FORCE IN THE BARREL AS A FUNCTION OF THE BULLET TRAVEL 

Military Ballistics  GM Moss, DW Leemines, CL Farrar 
Figure II3/3 
• The parameters influencing the speed of the bullet in the barrel
Let's move on to the study of what happens in the gun and see the parameters that act on the movement of the bullet.
The phenomena, especially those concerning the production of gases and their actions on the bullet, are complex. The combustion of the powder is not instantaneous. Pressure waves propagate from the breech or from the bottom of the case towards the base of the bullet, are reflected there, go backwards while meeting others that propagate forwards, thus creating a phenomenon of standing waves whose distances between the nodes and the pressure bellies vary as the bullet advances.
The problem can nevertheless be simplified in part by considering on the one hand that gases constitute a continuous medium, i.e. that an infinitesimal element (particle) of gas contains a large quantity of molecules, and on the other hand by making the hypothesis of a uniform gas density from the breech to the base of the bullet. The experiment shows that this approximation is suitable for speeds up to 1000 m/s and we will see that it is useful when we want to have an idea of the kinetic energy of the gases at the exit of the gun.
Let's go back to our bullet and see to which solicitations it is subjected. These are forces. We have detailed the forces of friction. As we have seen, we add them together (vectorially) to obtain a resultant which we will name FR for force of resistance to the advance. It will be directed towards the rear and obviously opposed to the force FG due to the action, on the base, of the pressure generated by the gases, which is directed, it, towards the front. Our system of resultant forces could not be simpler: an FG force which tends to accelerate the bullet towards the front and another FR which will tend to slow it.
These two forces can, in turn, be added vectorially to give a resultant force applied to the bullet that we will call FP. Taking as positive the direction of the breech towards the muzzle of the weapon, we obtain a simple equation linking
FP, FR andFG :
Therefore, the above equation allows us to consider three cases:
• FG > FR : FP is positive. The bullet is accelerated. Its speed increases towards the muzzle of the gun ;
• FG = FG : FP is zero. As the bullet advances FG decreases. At a certain moment, very brief, we have the equality of the forces FR and FG. At this moment the acceleration is zero and the speed of the bullet is constant;
• FG < FR : FG continuing to decrease, if the barrel is long enough (too long), FR being greater than FG, the acceleration becomes negative. The bullet decelerates. It will leave the barrel with a lower velocity than it would have with a barrel of the right length. In the limit, the bullet would stop in the barrel.
The image below shows the simplified principle of a firearm with the three forces acting on the bullet.
SIMPLIFIED PRINCIPLE OF A FIREARME 

Figure II3/2 
The friction in the barrel is not constant. They are particularly important at the time of the taking of stripes and their engravings in the jacket of the bullet or the belt if it is a shell. However, once this difficult passage is done, one can admit, without harming the generality, the hypothesis that they are constant, or take an average value, for the rest of the path in the gun. At our level, to consider them variable would complicate the problem unnecessarily and would not help the understanding of the phenomenon. Starting from these premises, we will determine the parameters which have an influence on the speed of the bullet once the rifling has been taken. Starting from the equation that we have just established linking FP, FR and FG, the calculations are not long and we will be able to detail them below.
When we start this kind of enterprise, we must know what we are looking for and where we want to go. In our case, we want to arrive at a relationship where the place of each parameter will allow us to understand how it acts on the exit velocity of the bullet. At this level of study, it does not matter to us how finely this parameter varies. The important thing is to understand how it acts globally. We therefore place ourselves in the perspective of a qualitative analysis. The quantitative aspect will eventually follow but, in this case, we will have to carry out measurements whose results will be integrated into the equations.
Simplifying assumptions
These hypotheses will allow, without detracting from the generality, to arrive at a relatively synthetic relationship concerning the parameters influencing the exit velocity of the bullet.
Hypothesis  1  Friction
We have seen that our study starts once the scratches have been taken and that from this moment we consider the friction as constant. We decide on this choice because, whether these frictions are constant or not, they slow down the bullet.
Hypothesis  2  The pressure in the barrel
As we have seen, the phenomenon is complex. We have already admitted the hypothesis that the gases constitute a continuous medium and have a uniform density from the breech to the base of the bullet. The force FG created by the action of the gases on the base of the bullet is of the form :


Product of a surface by a pressure 
Avec
L = length of the barrel ;
mp = mass of the bullet ;
Vp= velocity of the bullet ;
Pp = pressure at the base of the bullet ;
A = crosssection of the bullet base ;
xP = position of the bullet ;
PGmoy = average gas pressure ;
FGmoy = average gas force ;
FP = resultant force applied to the bullet ;
FG = orce due to the action of the gases on the base of the bullet ;
FR = force of resistance to the advance due to friction.
We do not know how the intensity of FG varies with time. What we do know, however, is that it pushes the bullet. How it pushes it, only pressure measurements would tell us, but in a global way, it pushes it. From then on, we can consider an average pressure in the barrel which is given to us by knowing the exit speed of the bullet and the length of the barrel. Physics gives us an equivalence relation between the kinetic energy of the bullet and the work of the average force of the gas:
The above relation has the advantage that the average force* FGmoy can be really calculated as soon as the mass of the bullet, its muzzle velocity and the length of the barrel are known.
* We must be careful that introducing an average force implies de facto an average pressure. It is important to understand that these assumptions have only one purpose: to understand the phenomenon. These average values should not be used in the calculation of the strength of a weapon. Indeed, the averages tend to smooth the curves and to "plane" the peaks. The graphs above (Graphs 1 and 2) speak for themselves: the pressure peaks are high and it is from these that the resistance of the weapon must be defined.
It is useful to show the average force FGmoy as a function of pressure:

or with the average pressure 

we obtain 

Starting from the equation linking FP, FR et FG : 


The second law of dynamics allows us to write : 

By performing the following change of variable : 

we obtain 

Given that 

The last equation can be written : 

By integrating : 

we obtain 

and finally 

The last equation allows us to obtain Vp

Équ : II3/1 
Or, if we want to show FGmoy :

Équ : II3/2 
This equation is the result of mathematical calculations. We must now use it in the context of physics, of ballistics. It is important to understand that it gives us information concerning the influence of certain parameters on the speed of the bullet at the muzzle of the weapon. It should not be considered as a function giving the variation of the speed of the bullet in the gun. Moreover, neither time nor distance appear in the equation. To put it simply, we should not ask too much of it. We can see its limits quite quickly:
• If FG > FR , all is well, we are in a context of normal operation of a weapon and we have a value of Vp ;
• If FG = FR, nothing abnormal ballistically since in this case FP is zero and Vp is zero ;
• The difficulty arises when we consider the case whereFG < FR. This is particularly the case when, in barrels that are too long for the power of the ammunition, the speed of the bullet passes through a maximum and then decreases. This phenomenon can be observed on some defensive bullet launchers available in "long barrel(s)" and "short barrel(s)" versions. In this case, our equation sends us into an imaginary world mathematically speaking. Indeed, in this case the term under the radical is negative and there is no solution in the set of real numbers. The domain of existence of our relation starts at zero and, with these preliminaries, we can see what information it gives us. But first let's see a simplification of this relation.
Simplification of the relationship
We can avoid this problem of the existence of a solution to our relation by freeing ourselves from friction. In our reasoning we assume that the force FG due to the pressure is greater than the friction. We thus arrive at a simplified relation of the following form, close to the first one :

Équ : II3/3 
If we analyze equation 2 or 3 mathematically, we can deduce that the higher the average pressure, the larger the crosssection of the base of the bullet and the longer the barrel, the higher the velocity of the bullet at the exit of the barrel. Conversely, the mass of the bullet acts in the opposite direction, i.e., with all the other parameters fixed, the higher the mass of the bullet, the lower the muzzle velocity.
If this equation is interesting for the understanding of the phenomenon, it is nevertheless necessary, as we have already said, to put it in the physical context, to define its limits in the ballistic sense and to analyze each of the parameters.
We will immediately settle the fate of the section of the base A and the mass of the bullet mp : these are constant values. Let us see the others.
The average pressure P cannot be constant. Even in the adiabatic regime (no exchange with the outside) the volume at the rear of the bullet grows as the latter moves forward and the pressure necessarily decreases, once all the powder is burnt (Graph II3/2).
The length of the barrel L cannot be as great as one wants. There is an optimum length beyond which the force, generated by the gas pressure, which accelerates the bullet, becomes weaker than the frictional forces which tend to slow it down. Beyond a certain length of the barrel, the speed of the bullet decreases. It can even become zero, the bullet remaining in the barrel. This is why it is important to use the right powder liveliness according to the length of the barrel. One thus avoids a waste of energy (all the powder burns before the exit of the bullet) and, at the same time, the bullet leaves with its maximum speed (Graph II3/2).
• Undercalibrated bullets
Let's go back to the section of the base A and the mass of the bullet mp. We can see that, all the other parameters being fixed, if we increase the crosssection of the base of the bullet while modifying its mass little or not at all, we obtain a higher muzzle velocity. This is the principle of subcalibre bullets: the bullet is enclosed in a sabot made of a light material but with a large crosssection, and the whole thing is ejected at a much higher speed than the bullet would have reached on its own, if it had been fired in a gun of its calibre.
• Moment of inertia of the bullet and equivalent mass
Until now, we have considered the action of the FG force due to gases as advancing the bullet towards the muzzle of the weapon by fighting against friction. A more detailed analysis shows us that the FG force must actually fight against three oppositions :
1  Frictional forces ;
1  The force of inertia of the bullet whose mass opposes any modification of its translational motion ;
3  The force of inertia of the bullet due to its moment of inertia which opposes its rotation.
We combine the two forces of inertia (displacement and rotation) by introducing an equivalent mass μ :

with 

We obtain the force equation :

with "a" the acceleration of the bullet 
• Projectile rotation speed
On its trajectory the stability of the bullet is ensured by gyroscopic effect. It is spinning in the barrel by the rifling.
The speed of rotation ω at the muzzle of the weapon can be obtained using two formulas. One or the other is used depending on whether the pitch of the rifling or its angle α to the bore axis is known.

or 

Let us specify that the acceleration of rotation exists as soon as the scratches are taken. It is not necessary that the bullet has covered a distance equivalent to one step of the rifling to have its definitive speed of rotation. Moreover, in the formulas above, the length of the barrel does not intervene. Clearly, two bullets having the same muzzle velocity have the same rotation speed whether they are fired into a 2 inch or 4 inch barrel as long as the two barrels of different length are rifled at the same pitch, for example 25 cm.
• Momentum of the bullet and gases at the weapon's mouth
The momentum of the bullet and the gases at the muzzle of the weapon are the two main factors involved in the recoil phenomenon of the weapon. When we talk about these quantities of movement "at the muzzle", we mean just before the bullet leaves the barrel. In this case, the part of the gas stream at the base of the bullet has the same speed as the latter.
 Momentum of the bullet
Product of the mass of the bullet Mproj by its velocity Vprojj at the mouth, it is easy to calculate when we know these values.
 Momentum of gases
The problem is more delicate. Contrary to the calculation of the quantity of movement of the bullet which is easy because we know, by measurement, its speed at the mouth of the weapon, we cannot measure the speed of the gas stream. We will deduce it from that of the bullet.
If we want a calculation that is not too complicated, we must consider some simplifying assumptions. Indeed, an analysis of what happens in the gun behind the bullet leads us to the following observation. We keep the simplifying hypothesis of the homogeneity of the gas stream and we cut it, perpendicularly to the axis of the gun, into very thin slices ; we split this gas stream into slices as thin as we want and we see what happens. It is clear that the slice of gas in contact with the breech of the bullet has the same speed as the latter, but the slice of gas in contact with the breech or the bottom of the case has a zero speed. Between these two extreme slices, all the others have a different speed according to their position in the barrel. The problem thus seems complicated.
We solve this problem by considering not the velocity of each of the gas slices but that of the center of gravity of the gas stream. The position of the center of gravity of the gas stream must be known since it is its variation which gives its velocity. The position of the center of gravity of the gases depends on the distribution of their volume, which leads us to consider two cases: the case of the guns with chamber and the case of the guns without chamber.
In interior ballistics a distinction is made between two types of guns :
1  Barrels with a chamber : the end of the barrel containing the powder, the cartridge, is of a diameter greater than that of the barrel. This is generally the case with powerful weapons, notably most rifles, but also with certain automatic pistols firing what is commonly called "bottleneck cartridge ";
2  Chamberless barrels : the end of the barrel containing the powder, the cartridge does not have a diameter significantly different from that of the barrel. This is the case for example with revolvers and some automatic pistols such as those firing 9x19 mm ammunition.
Let's start with the simplest.
1  Chamberless barrels
A barrel without a chamber can be considered as a cylinder. The volume distribution being symmetrical with respect to the middle of the cylinder, when the bullet reaches the muzzle of the barrel, the center of gravity is in the middle of the barrel.
The calculation is not very complex and we present it below.
Noting :
Qg = gas momentum ;
xg = position of the center of gravity of the gas ;
x = position of the base of the bullet ;
Vxg = speed of the center of gravity of the gas at the point xg
;
mg = mass of the gas ;
ρ = density of the gas ;
A = area of the base of the bullet ;
VP = velocity of the bullet ;
QP = bullet momentum.
The momentum of gases is given by the relation : Qg = mg x Vxg.
Introducing ρ the density of gases and considering the variation of the position of their center of gravity, we obtain:
In chamberless barrels, the quantity of movement of the gases at the muzzle is equal to the product of the mass of the gases by half the speed of the bullet.
Intuitive method
We can consider an intuitive reasoning to arrive at the same result.
Let's replace the gases by a spring as propellant. The coil of the spring in contact with the base of the case is animated by the same speed. The velocity of the coil at the bottom of the firing bowl is zero. At a given moment, the speed of each of the coils increases from the breech to the base of the bullet. When the latter reaches the muzzle of the weapon, the middle coil, where the center of gravity of the spring is located, is considered to be well balanced and has covered half the distance covered by the base of the bullet, and its speed will also be half that of the bullet (see diagram below).
GAS MOMENTUM (Alternative thinking) 

2  Case of barrels with chamber
In barrels with a chamber, the distribution of the gas volume is not symmetrical due to the very presence of the chamber. When the bullet reaches the muzzle of the weapon, the center of gravity of the gases is behind the middle of the barrel. It has travelled less than half the distance covered by the bullet. It therefore has a speed of less than half that of the bullet.
There is no general formula giving the velocity of the gases when the bullet reaches the end of the barrel since the distribution of the gas volume is specific to each weapon. However, measurements carried out on various weapons make it possible to give orders of magnitude : let VCG be the velocity of the center of gravity of the gases when the base of the bullet reaches the muzzle of the weapon and VP the velocity of the bullet at the same moment, VCG is of the order of 0.46 to 0.47 VP instead of 0.5 VP in barrels without a chamber. The difference is minimal but it exists.
• Kinetic energy of gases
The table below gives the energy balance classically used during a shooting.
THE APPROXIMATE DISTRIBUTION OF LIBERATED ENERGY 

Military Ballistics  GM Moss, DW Leemines, CL Farrar 
It can be seen that the amount of energy distributed in the gases, either as heat or as kinetic energy, is significant.
It can be useful to know the kinetic energy of the gases at the mouth for various reasons, for example to have an idea of their lesion potential. It is known that a close contact shot to the skull with a blank bullet is likely to result in serious injury or death.
To obtain the kinetic energy of the gases at the muzzle of the gun, it is classic to start from the Lagrangian gas continuity equation, which allows one to know the distribution of the speed of the gases behind the bullet :
We assume a uniform density of gases from the breech to the base of the bullet :
Integration of the Lagrange equation gives a result of the form :
Kinetic energy of gases : 


If this relation presents a sympathetic aspect, it is not completely right. Indeed, beyond 600 m/s the density of gases begins to be no longer uniform. We have to introduce a coefficient δ in order to obtain the following expression:
Kinetic energy of gases : 


Up to a speed of 600 m/s, we take δ = 3. Beyond that, we refer to graph 4 below:
VARIATION OF δ AS A FUNCTION OF BULLET VELOCITY 

Interior Ballistics of Guns  M. Krier, M. Summerfield 
It can be seen that up to 1000 m/s, δ varies slightly and that for conventional small arms, the gas velocity at the rear of the bullet is about onethird that of the bullet.
The mathematical solution of the problem of the kinetic energy of the gases at the weapon's mouth is relatively long. The interested reader can find here the calculations.
• The recoil of the gun
We cannot close the chapter on domestic ballistics without addressing the phenomenon of gun recoil.
It is accepted that gun recoil is due to three components :
1  Momentum of the bullet ;
2  Momentum of gases ;
3  Muzzle blast.
Points 1 and 2 have just been discussed and allow us to pose the following equation:
The sign "" in front of the second member indicates that the sum of the quantities of movement of the bullet and the gases is in the opposite direction to that of the weapon and that the sum of all the quantities of movement is zero, which it was for the system before the shot was fired.
Point 3 is more difficult and deserves our attention because experiments show that the contribution of the impulse due to the blast of the gases released at the muzzle can reach nearly 30% on certain powerful weapons such as war rifles in 7.62 x51 mm calibre. This is the reason why muzzle brakes are so useful.
We can try to solve this problem by looking at the functioning of the nozzles which has been well understood for a long time. We can thus say that the thrust exerted at each moment on the weapon by the ejected gases is given by the equation :
Thrust = Net Gas Pressure at Exit x Exit Area x Mass Flow 
Even before studying this equation briefly, we must be suspicious because this relationship describes the operation of a nozzle in a steady state, whereas the phenomenon we are interested in is transient.
The application of this equation to our particular problem of weapon recoil leads us to the negative conclusions reached by other ballistic experts interested in this phenomenon :
The Net Gas Pressure at Exit is simply the average gas pressure at the muzzle minus the atmospheric pressure. The pressure in the barrel before the bullet exits is several hundred bars. It drops to zero in a fraction of a second, this rate of change, necessary for our study, is unknown and all the figures that we could use could only be conjectures.
The Area of the Exit, which is the cross section at the mouth of the gun is easily calculated from the diameter, known, of the gun.
The Mass Flow is, like the Net Gas Pressure at Exit, unknown as far as our recoil calculations are concerned.
Finally, all this equation tells us is that the higher the atmospheric pressure, the lower the recoil of the weapon due to the muzzle blast, and that the higher the atmospheric pressure, the lower the Net Gas Pressure at the exit is. Indeed, it is easy to understand that if the ambient air is at the same pressure as the gases inside the gun, the gases will not come out at all and the recoil due to this effect will be zero.
The above equation does not bring us much information about our problem. Let's turn to the literature.
The specialized books give divergent values of gas exit, going from 670 m/s (Kneubuehl) to 1500 m/s for other authors (Engineer General Moreau). In their defense, these values depend on the caliber, the type of powder and the power of the weapon. Only Kneubuehl specifies that the value given concerns the 9 mm Luger and that for the .38 Special the speed of exit of gases is 930 m/s.
A calculation carried out from the half angle at the top of the Mach cone at the rear of a bullet (gas/bullet nose interaction) at the exit of the barrel, the average speed of the gas stream and the speed of sound in the gas volume at a temperature of 1200 K, gives us an initial gas ejection speed of 1013 m/s for a bullet with a muzzle velocity of 330 m/s. Taking into account the uncertainty of the measurements this calculated value is to be taken as an order of magnitude.
It seems that on this subject the theory shows its limits and that it is towards experimentation that one must turn.
By measuring the speed of recoil of the weapon Vgun during a shot and knowing its mass Mgun, we easily obtain its quantity of movement Qgun. It is the same for the quantities of movement of the bullet Qproj and of the gases Qgas. If we call Igas the impulse due to the gas, its value is given to us by solving the equation below :
Igas = Qgun  (Qproj + Qgas) 
We stayed for a while at the mouth of the weapon. Let's move forward a little and observe the exit of the bullet from the barrel. We enter the field of intermediate ballistics.
But before going any further we are in a situation where we have a bullet which, through the transformation of a chemical potential energy into a kinetic energy, is endowed with a capacity to perform mechanical work on a target. We know that it is also animated by a quantity of movement capable of being transmitted violently. Which of the two criteria should be chosen to evaluate its efficiency: kinetic energy or momentum*? The diagram below gives us some guidelines.
KINETIC ENERGY VERSUS MOMENTUM There is conservation of kinetic energy and momentum, but... 

Note : Only in the ideal case of a perfectly elastic shock (no loss of energy) does "kinetic energy > kinetic energy". Hence the need to know how to choose the right tool at the right time and according to the phenomenon.
*Answer : the kinetic energy of a bullet gives it the capacity to do mechanical work on a target. This mechanical work will be all the more important as it will be well used. It is therefore the kinetic energy that is the criterion for the effectiveness of a bullet. Even if many theories have been developed on the stopping power, kinetic energy remains the only truly scientific criterion used in ballistics laboratories.
However, we will see that, in some cases, momentum also has a say.
III  INTERMEDIATE BALLISTICS
A bullet in the storm
It should be noted that some ballistic experts, and not the least, make no distinction and do not speak of intermediate ballistics. They consider that, as soon as the bullet leaves the barrel, it is a question of external ballistics and that what happens at the muzzle of the weapon is an epiphenomenon that emphasizes the need for good stabilization, which is true in the literal sense of the term. We do not advocate any particular method but, considering the interesting phenomena that take place there, we have decided to treat this initial phase of external ballistics separately.
The intermediate ballistics corresponds to the initial phase of the flight of the bullet during which the gases still exert an action on the latter.
During its path in the barrel, the bullet plays the role of a piston and the speed of the gases behind it is limited by its own speed, provided that there is a good seal between the bullet and the inner wall of the barrel. This is not always the case in practice. It often happens, in fact, that some of the combustion gases precede the bullet at the muzzle of the barrel, as shown in the video* below.
The sequence is detailed in the image below.
IMAGE SEQUENCE OF GASES AT THE MUZZLE OF BERETTA 92F 

At the exit of the gun, the gases relax and, no longer blocked by the base of the bullet, accelerate and overtake it.
At the very beginning of this phase, the difference in speed between the bullet and the gases is such that a shock wave is created at the base. This phenomenon is very clear in high speed shooting, as shown below, the video* and the sequence of images.
The sequence is detailed in the image below.
IMAGE SEQUENCE OF GASES AT THE MUZZLE OF RUGER SP 101 

We are in the same situation as if the bullet was moving in the gas with the base forward and at a supersonic speed.
The bullet is subjected to two major constraints :
1  The brutal encounter with the ambient air which is called initial percussion ;
2  The gases that exert a thrust on the base.
When the bullet leaves the barrel, its longitudinal axis is never perfectly aligned with that of the barrel. Due to the combined actions of the initial percussion and the gases, it is subjected to a couple of forces that tend to make it tip over.
The diagram below represents the situation.
GASES AT THE BULLET BASE 

• The consequences of the action of gases on the bullet
Below, we see the normal behavior of a bullet designed to be stable in relation to the medium, the air, with which it interacts.
NORMAL BEHAVIOR OF A BULLET 

The following image shows us the situation in which the bullet is subjected to the action of gases.
EQUIVALENT BEHAVIOR OF THE BULLET IN GASES 

It is during this phase that the effectiveness of the stabilization is crucial.
The phenomena at the mouth of the weapon are schematized below.
SCHEMATIC PRESENTATION OF THE PHENOMENA AT THE MUZZLE 

The stabilization of the bullet having been effective, it leaves this agitated region to approach the air phase of its trajectory. We enter the external ballistics.
IV  EXTERNAL BALLISTICS
• The two approaches to external ballistics
Classically, the study of external ballistics is subdivided into two parts :
1  Ballistics in vacuum : the only force acting on the bullet is its weight ;
2  Ballistics in the air : we take into account the interaction of the bullet with the medium in which it propagates, the air.
IV1  BALLISTICS IN VACUUM
The bullet is subject to only one force : its weight.
It is animated by a velocity whose representative vector, V, tangent to its trajectory, generates a plane, the shooting plane.
This velocity vector V can be decomposed into a horizontal and a vertical component:
The horizontal component along the xaxis:

Équ : IV1/1 
The vertical component along the yaxis:

Équ : IV1/2 
With α the angle of fire.

Figure IV1/1 
• Forces applied to the bullet
We have decomposed the motion of the bullet into two components along the x and y axes. To study this motion on the trajectory, we will define the forces that apply on these components.
Depending on x :

Équ : IV1/3 
Depending on y:

Équ : IV1/4 
With, for the equations : 

We remind the equations of uniformly accelerated motion since, in our case, this is what it is all about :

Table IV1/1 
• Interpretation of the force equations depending on x and y
Depending on x, the force Fx is zero so the acceleration is zero, the change in motion is also zero. There is no air or any other friction to modify the motion in this direction. It could continue indefinitely, identical to itself, as long as the bullet does not encounter an obstacle.
Depending on y, the force Fy is not zero. As for the horizontal component, there is no air and no friction. But there is the weight of the bullet which is a force. It acts on the vertical component of motion. We can also say that it acts differently according to the phases of the trajectory. It slows down during the ascent, so much so that at a certain moment the vertical speed is cancelled, then accelerates the movement during the descent in such a way that, for the same altitude, the speed of the bullet is identical during the ascent and the descent, but in the opposite direction. This explains why, when shooting on horizontal ground, the arrival speed is the same as the initial speed, both in its intensity (its modulus) and in the value of its angle.
The only force modifying the movement along the vertical is the weight of the bullet which is the product of its mass and the acceleration of gravity g. We consider that g does not vary on the trajectory. This is a good approximation if one is not an artilleryman and if one does not send bullets in curved shots several tens of kilometers away. We are therefore, in the vertical direction, in the case of a uniformly accelerated movement.
All the initial parameters such as the speed Vo, the angle of fire, the weight of the bullet (considering that g does not vary on the trajectory) are fixed at the start of the shot. They will not vary during the flight of the bullet. We can represent the trajectory by a generalized equation.
• Generalized equation y = f(x)
Table IV1/1 gives us the variations of y as a function of time. It is the result of the application of the second law of dynamics F = m x dv/dt which, by double integration gives us y = f(t). Following the same process, the table also gives us x = f(t). We have :


Équ : IV1/5 
et 


Équ : IV1/6 
Equations IV1/5 and 6 are the general equations that we will adapt to our situation and simplify at the same time. The acceleration a is replaced by the acceleration of gravity g. We have seen that g has an action only on y. We simplify the equations by taking x0 = y0= 0, that is to say that we study the motion from the mouth of the weapon whose coordinates are x0 = 0 and y0 = 0. We obtain:


Équ / IV1/7 



Équ : IV1/8 

The sign "" means that g, a vector quantity, is opposite to the direction of motion.
At this time we have information on the values of x and y as a function of time t. Very often, it is useful to know the trajectory as a function of the distance x, i.e. in the form of a function of the type y = f(x) which we will obtain thanks to a simple transformation.
En posant t = x / Vcosα et en injectant cette valeur à la place de t dans l'équation de y, nous obtenons, après quelques arrangements, l'équation générale de y en fonction de x.
By positingt = x / Vcosα and injecting this value in place of t in the equation of y, we obtain, after some arrangements, the general equation of y as a function of x.

Équ : IV1/9 
This is the equation of a parabola. By reducing the bullet to its center of gravity, we can draw a curve of its trajectory.
PHASES OF FLIGHT OF A PROJECTIL IN THE VACCUUM Parabolic trajectory 

Figure IV1/1 
For the clarity of the diagram IV1/1, the bullet is reduced to its center of gravity G.
The only force to which the bullet is subjected is its weight W.
The horizontal component Vx of the velocity V is constant. Only the vertical component of V, Vy varies.
V varies because Vy varies. Under the action of W, Vy decreases in ascending phase (phase 1 and 1').
Vy is zero at the apogee of the trajectory and V is minimum (phase 2).
During the descending phase (phase 3), under the action of W, Vy increases, as does V.
At the same altitude, the value of V is identical in modulus but of opposite direction.
The final speed is equal to the initial speed.
Continuing our study, we will determine on this parabola a certain number of characteristic points. Cf. diagram IV1/2, below :
FIRING IN THE VACUUM Firing parabola Characteristic points 

Diagram IV1/2 
• The parabolic path
Figure IV1/2 above shows a parabolic path and its particular points giving the distance and time to reach them.
The line (D) is the director of the path parabola and the point F its focus.
The circle with center O and radius A is the locus of the foci of all path parabolas for the same initial velocity Vo and variable shooting angle α.
The tangent (T) is the bisector of the angle AOF.
The point P is the characteristic point of the envelope of the shooting parabolas under variable shooting angle and constant Vo . It is the parabola having for focus the origin and for vertex the point A.
For a given initial speed Vo, all the points of the plane under the envelope of the shooting parabolas can be reached by two shooting parabolas corresponding to two shooting angles.
The points of the plane outside the envelope of the shooting parabolas cannot be reached. This is why the envelope of the shooting parabolas is called the safety parabola.
• Firing parabolas
On the diagram IV1/3 below appear firing parabolas for different angles of firing for the same initial speed. One notes, in particular, that the targets, T1, T2, T3 and T4 can be reached by two parabolas having different firing angles. In a general way any point of the plane under the safety parabola can be reached by two firing parabolas. This can be of practical interest, as the same target can be reached by a relatively low parabolic trajectory or by another one whose higher apogee allows, for example, to pass over an obstacle. The red curve represents the maximum range with a 45° firing angle.
FIRING PARABOLAS 

Diagram IV1/3 
• Define two shooting parabolas for the same target with the same initial speed  Geometric method
On the diagram IV1/4 below, we propose to reach the target (red point)T, which is on a first shooting parabola (blue), with another shooting parabola (green).
The start of the shot is at 0. The focus F1 of the first parabola (Shot 1) is on the circle C1 centered at 0 and tangent to the directing line (D).
The target T is, by definition, at equal distance from F1 and from (D). It is thus the center of a circle C2 tangent to (D).
The two circles C1 and C2 intersect at F1, focus of the first parabola (Shot 1) and at F2, focus of the new parabola (Shot 2). The tangent (T2) is bisector of the angle A0F2, which allows us to find the new angle of fire.
HOW TO DEFINE TWO SHOOTING PARABOLAS 

Diagram IV1/4 
IV2  BALLISTICS IN THE AIR
• The problems of external ballistics
During its flight path, the bullet is subjected to a certain number of constraints which give rise to a system of forces. Some of these forces have a strong influence on the trajectory while others have a lesser effect. In the case of relatively short trajectories, such as those of smallcalibre bullets, some of these effects are nonexistent and we can ignore them, while others, because of the reduced flight time, do not have the time in practice to act in a significant way. The constraints to which the bullet is subjected have been classified into two categories : the main problem and the secondary problem.
• The classification of problems in external ballistics
The forces acting on the bullet have been classified according to the importance of their influence on the trajectory.
1  The main problem :
 The gravity supposed constant in intensity and direction (weight of the bullet) ;
 The tangential resistance of the air.
2  The secondary problem :
 The Earth(sphericity, rotation);
 Gravity (variation with altitude and longitude, convergence of verticals) ;
 The atmosphere (variation of density with altitude, atmospheric wind) ;
 The bullet (gyroscopic drift).
As we are particularly interested in the ballistics of small arms, we will limit ourselves mainly to the study of the main problem. The study of the gyroscopic drift will nevertheless be of interest in order to determine its influence on long range shots.
• The main problem
During the whole phase of its flight, the bullet is subjected mainly to two forces: the force of gravitation which makes it fall towards the center of the Earth and the drag force, the retardation, due to the air in which it moves, which slows it down and prevents it from going as far as if it were shot in a vacuum.
We have seen previously that the first bullets were spherical. Very quickly, it was realized that this shape was not the best in terms of aerodynamics and precision. Therefore, the need to manufacture more streamlined bullets became apparent.
However, in order for this to work, the bullet must move with its nose forward and its longitudinal axis must be tangent to its trajectory, which is not necessarily obvious.
• Trajectory and range
If we were to shoot in a vacuum, that is to say if there were no atmosphere, our bullet would describe, as we have seen, a parabola. In reality, it would describe an ellipse whose one of the foci would be the center of the Earth. It would describe a circle around the center of the Earth for a precise speed but, taking into account its initial speed, in practice too weak, it would end up meeting the surface of our planet.
Its range would be much greater than it is in the air. A war rifle bullet fired at an angle of 45 degrees (maximum range in a vacuum) would hit the ground several tens of kilometers away.
To get an idea of the shape of a firearm bullet's trajectory, let's look at a golfer's drive or a soccer player's shot. It starts out straight, then quickly curves and falls to the ground very steeply.
The trajectory of a bullet is much flatter, but the shape is the same.
Figure IV2/1, below, gives an idea of the difference between a shot in a vacuum (curve A) and a shot in the air (curve B). If it is the same bullet, (the two curves on the graph are not to scale), the trajectory A is about ten times shorter than that described in B.

Diagram IV2/1 
IV3  DRAG SLOWING  RETARDATION
The bullet is slowed down by the air in which it propagates. This braking, called "retardation", depends on many parameters such as: the mass of the bullet, its maximum diameter or master torque, its shape.
IV31  BRIEF HISTORY
The laws of slowing have evolved with time, the increase in the speed of the bullets and the better understanding of aerodynamics.
Historically, the experiment had highlighted three laws :
 The air resistance is proportional to its density;
 The air resistance is proportional to the cross section of the bullet ;
 The air resistance R, for bullets of the same section and of minor different shapes, can be put in the form R=iF(v) where i, called form factor (we will come back to this), is a coefficient independent of the speed and F(v) the same function of the speed for all the bullets.
Given that 

Introducing 

in F(v), we have 

Introducing 

, c is the ballistic coefficient of the bullet. 
The function F is an acceleration, that of the bullet, whose coefficient is equal to 1.
A non negligible phenomenon had to be taken into account beforehand: the air resistance varies with the speed, but in a way that is not very consistent with our daily experience.
• Air resistance... not always proportional to the square of the speed
We often hear: "The air resistance on a mobile is proportional to the square of their relative speed".
This is true in a range of speeds that corresponds to our daily experience: the speed of a classic car, HSR, Formula 1 at full speed. We will see that the increase in the speed of bullets has led ballistic experts to reconsider their first equations.
• The variation of the air resistance with the velocity
Research in ballistics was fertile in many countries. France has acquired an international reputation in this field. It is rare to find a foreign work on ballistics which does not refer to the French work in particular through the law of Gavre. We will present later details on the French work, but for the moment let us remain in the historical generalities.
• The consequences of increased velocity
The increase of the bullet speed led to compare F(v) to monomial functions v2, v3, v4 or even v5 or to analytical functions of the type F(v) = a + bv until 350 m/s.
Below is a graph showing the variation of braking as a function of speed. Note its sudden variation between 300 and 400 m/s per second corresponding to the region of transonic speeds illustrating well this notion of "sound barrier".
Variation of F(v) as a function of v 

Cours de Balistique Extérieure – M. L. Besse

Interpretation of the graph
The above graph is old but is in every way similar to the modern curves representing drag coefficients that we will discuss later. It gives some interesting information. First of all, the problems of stabilization of the bullet are particularly important in the vicinity of the speed of sound. This is an area to be avoided, if possible. This means that it is desirable to launch the bullet at a speed much higher than the speed of sound and to avoid, if possible, that on its trajectory it enters the transonic zone before having reached the target. Another alternative, but reserved for short distance shots, is to choose the subsonic region, that is to say a speed clearly lower than that of sound.
Once the "sound barrier" is passed, everything seems calmer. The variation of the drag is smoother, one could almost consider it as constant in quite large speed ranges. Hence the idea of the calculation by arcs. We consider the braking coefficient constant over a range of speeds, the speed at the exit of this range corresponds to the initial speed of the following range of speeds and so on. We join the portions of the trajectories one after the other and we obtain the complete trajectory.
The graph also shows something more subtle which highlights the weakness of the old equations. In the latter, the speed of the bullet is involved. The braking force depends on it. But if we look more closely at the position of the sound barrier, as its name indicates, its position depends on the speed of sound. If the speed of sound "moves" for variable but very real reasons, the sound barrier moves at the same time, dragging the whole curve to the right or to the left. In future equations, the speed of sound, which is not a constant, will have to be taken into account in one way or another.
But let us not anticipate, let us continue to be interested in the evolution of knowledge.
The laws of physics, therefore of ballistics, being the same for everyone and perhaps because of a certain porosity between laboratories of various countries, close results, not to say identical, have been found.
• The Mayevski's coefficients
A Russian ballistician, the colonel MAYEVSKI leaned, him also, on the problem and established a table of the exponent of the speed which is constant only for ranges of given speeds.
When the speed increases "n" increases, goes to three, four and then five when we reach the speed of sound. It then decreases sharply. This means that the air resistance reaches a maximum in the vicinity of the speed of sound (transonic speed). It is probably for this reason that the term "sound barrier" was born, inspired by the difficulties encountered by the first supersonic aircraft manufacturers and especially their test pilots.
Below is a table of Mayevski's coefficients.
Note : Some books give n = 2 for v between 50 and 240 m/s. It is traditional to consider that the air resistance is proportional to v (n = 1) for low speeds.
• The retardation
The first experiments whose aim was to define a global equation usable whatever the type of fired bullet ended in failure. It was thus defined a typical bullet whose mass and caliber were taken as a unit for all the calculations, well characterized in terms of its shape, on which the delay R could be put in a simple form :
In this formula, R represents the delay expressed in m/s/s, A a function depending on the characteristics of the bullet.
This equation was a good model as soon as one considered that the speed of sound was constant or that its variation had no appreciable incidence on the results of the shots. As we have seen, it was necessary to evolve.
IV32  TOWARDS A NEW MODEL OF THE SLOWING FORCE
The evolution of weaponry techniques has made it possible to send bullets at increasingly high speeds. It was realized that a single formula of delay could not be used, simply, to account for the braking of the bullet for all the ranges of speeds of which it is animated throughout its trajectory.
Moreover, the high speeds, close to the speed of sound, even supersonic, were not any more the reserved domain of the gunners. The jet propulsion allowed planes to reach these speeds and was going to induce the development of aerodynamics. One realized in particular that the delay or the drag force depended, in reality, not on the speed of the bullet but on the ratio between its speed and that of sound. This ratio is called Mach's number in homage to the physicist who discovered it.
Nowadays, the braking of the bullet is represented by a force, the aerodynamic drag, derived from the Bernoulli's equation:

with 

Before going further, it is interesting to stop for a moment on this formula.
The first member of the equation, the force FDrag entrained is expressed in Newtons. The sign "" in front of the second member indicates that the force opposes the motion of the bullet, ρ is the density of the air, V the speed of the bullet. We note that it intervenes with the square whatever is the speed of the bullet. Indeed, we are no longer in the application of the Mayevski's coefficients, the exponent must always be 2 at the risk of not being homogeneous to a force. S its apparent surface, CD the drag coefficient, dimensionless. The second member of the equation, expressed thus, has the dimension of a force.
• The drag coefficient CD
The drag coefficient CD is a function of several parameters :

with 

In practice, we can neglect the Reynolds' number when the bullet yaw is not impotant. In this case, the drag coefficient depends only on the Mach number, i.e. :
Below is an example of a representative drag coefficient curve for a 7.62 mm Nato bullet.
DRAG COEFFICIENT FOR A 7,62 mm NATO BULLET 

It should be noted that it is impossible to represent braking by a single generalized equation. It will be necessary to take into account the variation of CD with the speed. CD thus varies during the flight of the bullet.
• The drag coefficient varies with the shape of the bullet
The drag coefficient CD varies with the Mach number and the shape of the bullet.
THE DRAG COEFFICIENT VARIES WITH THE SHAPE OF THE BULLET 

• The ballistic coefficient and the form factor
1  The ballistic coefficient
In a classical way, we use the second law of mechanics: F = mdv/dt.
We introduce 

with 

Note that some works present the ballistic coefficient C as the inverse of that presented above. The mass goes down in the denominator and the diameter squared goes up in the numerator. In this case C is found in the numerator of the formula giving C*T.
2  The form factor
In step 1, we arrived at the result that the ballistic coefficient varies as a function of the mass and diameter or master torque of the bullet. This is an interesting result, but probably not sufficient. What about bullets of the same diameter, of the same mass but of different shape? The image below simply illustrates the question.
he P1, P2 and P3 bullets have different air penetration. This seems obvious to the naked eye and wind tunnel tests confirm it. The ballistic coefficient, as presented above, is insufficient ; the shape of the bullet must be taken into account.
We therefore introduce a form factor that we call "i" for the occasion. We thus obtain a more complete and definitive ballistic coefficient.
To be complete, we must go a little further.
• The generalized ballistic coefficient and the form factor
When we want to describe a phenomenon, we consider all the factors that have an influence on it, even if it means getting rid of some parameters that we may consider as negligible.
The generalized ballistic coefficient : 

with 

Form factor : 

donne 



ij is the average ratio of the drag coefficient of a given bullet to that of an experimentally determined normalized bullet.
We have chosen to present a little theory before going into more detail about what happens on the trajectory of the bullet. We are thus better equipped to study the forces to which it is subjected.
IV4  THE FORCES TO WHICH THE BULLET IS SUBJECT
The interaction of the bullet with the air is at the origin of three forces.
Note: generally, the influence of the Reynolds number can be neglected
• Yaw  Reference trihedron for the bullet
In practice the bullet is never perfectly aligned with the velocity vector. The angle αt between its axis and the velocity vector is called the obliquity.
• Six degrees of freedom
The study of the movements of the bullet leads us to define a reference trihedron on it.
SIX DEGREES OF FREEDOM  REFERENCE TRIHEDRON


Note : the direction of the rotations can be reversed according to the books. Modern ballistics merges pitch and yaw into a single term yaw.
• Total yaw αt
The total yaw of the bullet has a horizontal component called the angle of deflection, and a vertical component called the angle of incidence or angle of attack.
SIX DEGREES OF FREEDOM  REFERENCE TRIHEDRON ON TRAJECTORY 

We introduce δ such that:
• Aerodynamic forces and their resultant
By symmetry along the longitudinal axis, the resultant of the aerodynamic forces FR lies on this axis.
There is no symmetry along the transverse axis passing through the center of gravity G. The force FR is at a distance from this point. The forceFR and the weight P create a torque which induces a rotation.
EARODYNAMIC PRESSURE INDUCES TUMBLING PHENOMENON 

A yaw αt even very small at the exit of the gun produces a force FR itself very small but nevertheless sufficient to increase αt which, in turn will allow an increase in the intensity of FR. A rocking motion is initiated.
•Decomposition of the resultant FR of the aerodynamic forces
We treat FR in two stages.
1  First step. Reducing FR at the center of gravity
Reducing FR at the center of gravity gives an equipolent force to FR and a rotational moment :
2  Second time. Decomposition of FR
We decompose FR into two forces: drag FD and lift FL (L for "lift").
DECOMPOSITION OF FR INTO FD and FL 

• The drag forceFD
The drag force opposes the movement of the bullet and its value is :

with 

It is collinear to the velocity vector V and of opposite direction.
• The lift force FL
he force of lift is perpendicular to the motion of the bullet. On this subject, it may be useful to recall that, on most diagrams, we see the lift directed upwards. These diagrams should be taken as snapshots in which the lift, at that moment, is directed upwards. At another moment, it will be directed in another direction, possibly downwards due to precession. We must therefore be careful not to take an "instantaneous" as a generalization. Only in aeronautics is it ensured that the lift is always directed upwards.
The animation below shows the action of FL according to the angle of obliquity. FL is applied to the center of gravity and thus tends to move it, at each instant, in the direction of its action. Looking at the front of the bullet in the axis of the velocity vector, we would see the center of gravity, if it were under the influence only of FL, describe a circle, projection of a helical motion.
In the animation below, the red circle is the location of the successive positions of the center of gravity (red point). The green circle represents the precession due to yaw. FL is represented by the brown arrow.
Action of FL on the bullet
(Move the mouse over the image)


Remember that the animation above represents only the action of the FL force. Other forces will be added which will make the movement of the center of gravity more complex.
The value of the lift force is :


with 

We find that : 

If the axis of the bullet were perfectly aligned with the velocity vector, there would be no lift force. Only drag would exist.
The drawing below shows the instantaneous action of the two forces, drag FD and lift FL.
The aerodynamic forces cause the bullet to tilt. It is essential to stabilize it.
In the following chapter, two stabilization methods are presented: stabilization by tail and stabilization by gyroscopic effect. These two methods can be used on smallcalibre weapons. The stabilization by tail, because of the presence of the latter, is rather used on subcalibre bullets. The application of the gyroscopic effect is by far the most common method for stabilizing bullets.
IV5  STABILIZATION OF BULLETS
• By nature, the bullet is not necessarily stable
ntermediate ballistics has shown us that at the exit of the gun, the gases acting violently on the base of the bullet tend to destabilize it.
When it exits from the barrel, the bullet meets, at high speed, the immobile ambient air. It undergoes a shock which is called, in this case, initial percussion and which also tends to destabilize it.
Moreover, we have seen that, throughout its aerial path, the bullet undergoes a set of interactions with the air whose resultant has, because of the symmetry of revolution, its point of application on the longitudinal axis and in general far from the center of gravity. The result is a couple of forces that apply to make it tip over.
The drawing below shows an unstable bullet by nature.
TUMBLING DUE TO AERODYNAMIC PRESSURE 

• The stabilization methods
There are several ways to stabilize bullets. We will discuss two of them : stabilization by tailplane and stabilization by gyroscopic effect.
 Fin stabilization
We will treat it quickly, for information. Better than a long speech, the following diagrams describe well the method which consists, with an empennage, in sending the resultant of the aerodynamic forces far behind the center of gravity. The couple of forces thus created tends to maintain the projectile with the point in front.
A projectile equipped with its tail is stable by nature.
FIN STABILIZED PROJECTILE 

 Stabilization by gyroscopic effect
For many bullets, the stabilization method used is the gyroscopic effect.
The bullet is rotated at high speed (several thousand revolutions per second) along its longitudinal axis. The braking forces of the air, which would cause the bullet to tilt without the rotation, give it a precessional movement at most, provided that the speed of rotation is well chosen. The short duration impulses due to the gas thrust at the exit of the gun and to the initial percussion induce, them, a movement of nutation which is temporarily superimposed on the precession and which is damped relatively quickly.
The diagrams below illustrate the phenomenon.
Precession 

Precession and nutation 

Read more about gyroscopic stabilization
IV6  THE TRAJECTORY IN THE AIR
The bullet being stabilized, it can move according to a trajectory of which one will find a representation below.
Description of the scheme
Along the xaxis : Dx is the horizontal projection of D and R. These two projections have the same value since W has no action on the horizontal component. On this axis Dx and Vx decrease when x increases.
Along the yaxis : Ry is the vertical projection of the vector sum of D and W. In the upward phase, D and W contribute to the slowing. In the descending phase D and W act in opposite directions. W tends to accelerate the fall and D to slow it. The direction of R tends to bring the trajectory back towards a vertical until D and W are collinear. The fall speed is constant when D =  W.
From the study of the trajectories of the bullets were born some interesting principles.
• The SaintRobert's principle
The SaintRobert's principle is stated as follows: on a family of trajectory with constant initial velocity Vo and initial retardation, the angle of fire α being variable, the points of the same distance OA, OB, OC are characterized by equal values of the lowering (drop) AM, of the duration of the path and of the remaining velocity (see diagram below)
The interest of this principle lies in the fact that from a known trajectory, it is possible to calculate others with different angles of fire. The condition being that all other initial values are identical.
• The principle of the rigidity of the trajectory
If we apply the previous principle to a very tense trajectory, the drop is very small compared to the maximum range. The line O1 M1 merges with the line OA . To reach the point M, it will be enough to increase the angle of fire by ε. The trajectory (T2) is obtained by simple rotation of (T1) around the origin 0. Everything happens as if the initial trajectory (T1) was constituted by a rigid wire. Hence the name of the principle of the rigidity of the trajectory (see diagram below).
It should be noted that this principle applies to small angles of site.
The consequence of this principle is that if α is the firing angle to hit a target at distance X, the firing angle to hit a target at the same distance on the line of site of (low) inclination s will be a + s by the parallelogram method.
• The alterations of the trajectory
The trajectory of a bullet can be subject to alterations due to its interaction with the air. The consequences are a deviation of the trajectory of the bullet in the horizontal plane which no longer corresponds to the point of impact. The main alterations are :
 The deviation : The deviation : It is due to a side wind which applies to the bullet a force in the same direction thus deviating its trajectory in the horizontal plane. The effect of this side wind is all the more sensitive as the shooting distance is important;
 The Magnus effect : When a rotating body moves in a flow of air, its surface drags the air particles in an asymmetrical way: on one side, the rotation speed of the ball is in the same direction as the air particles and accelerates them, which leads to a decrease in pressure. On the other hand, the rotation speed of the surface is opposite to the air particles and slows them down. In this case, the pressure increases. This pressure difference on the opposite sides results in a force that deflects the ball's trajectory. This effect is well known in soccer, tennis, etc., when you want to give a spin to the ball. In the case of a spinstabilized bullet with a certain yaw, the Magnus effect deflects its trajectory to the side opposite to its direction of rotation.
The Magnus force is added to the system of forces (see drawing below).
Read more about the Magnus effect
 The drift : Rational mechanics tells us that a body in rotation and in translation tends to align its axis of rotation with the tangent to its trajectory. A bullet stabilized by rotation therefore undergoes, over its entire trajectory, a rotation along its transverse axis of an angle at least equal to twice the firing angle. This tendency to align itself on the tangent to the trajectory means that the bullet "runs after its tangent". It is therefore constantly behind the latter. The combination of the speed of rotation along its transverse axis with the speed of precession, at a given instant, causes a shift of the instantaneous center of precession in the horizontal plane. The bullet is thus presented with a lateral obliquity with respect to the mass of air which it meets. As a result, its trajectory is deviated in the horizontal plane on the side corresponding to its direction of rotation. The diversion is opposed to the Magnus effect and is often called, for this reason, counter Magnus effect. In the case of a shot at short distance or when the trajectory is very tense, with a shooting angle practically zero, there is no drift.
The drift force is added to the system of forces. It is opposite to the Magnus force. For this reason, it is called the counterdrift. (See drawing below).
Read more about the drift
IV7  THE PROPAGATION OF THE PROJECTILE IN THE AIR
On its trajectory, the bullet makes its way through the air. This results in remarkable phenomena that deserve to be studied.
• Stability and transonic velocity
The stability of a bullet depends on the resistance of the air. We understand that its passage in transonic speed will disturb it somewhat. In particular when its speed, initially supersonic, decreases with the distance to become subsonic and that, at the same time, its speed of rotation also decreased.
• Subsonic and supersonic bullets. Shock wave, Mach wave. Wake
When the molecules of the ambient air are locally shaken (sound source or other...), this shaking is propagated from close to the neighboring molecules. It propagates at a precise speed, which depends on the characteristics of the medium, called specific speed. It is the speed of sound in this medium, the air in our case (a # 340 m/s).
It should be noted that there is transfer of energy but not of matter.
 Still source
If the source (black point) of the shaking is immobile, the waves generated will be either circular or spherical depending on whether they propagate along two or three spatial dimensions. In all cases, they will be concentric.
During its flight, the tip of the bullet strikes air molecules (this is the reason why it is braked). This shaking will propagate, in the form of circular waves, in all directions and in particular in the direction of the progression of the bullet. 

 Subsonic velocity
he bullet moves at a velocity lower than the speed of sound (subsonic speed), the sound waves it has generated will move away from it indefinitely and in particular from its tip. 

 Sonic or transonic velocity
The bullet moves at the speed of sound. The waves emitted by the shaking of the air molecules by its point remain at the level of the latter. The accumulation of these waves locally increases the density of the air and constitutes a kind of barrier that the bullet will have difficulty in crossing.
The front of the bullet will be accompanied by a wave front. 


 Supersonic velocity
The bullet velocity is greater than the speed of sound. The waves emitted at each moment are overtaken by the tip of the bullet and are left behind. This wave is, at the tip of the bullet, supersonic. It is a shock wave.
At the back of the bullet, the sound waves propagate at the speed of sound and interfere in places. It forms an accumulation of energy on the envelope pressing on the edges of the waves. This envelope has the shape of a cone whose top coincides with the tip of the bullet. This wake is the Mach wave. It is this dry "slam" that we hear that is clearly different from the mouth wave.


• Shock wave
A shock wave is a sudden variation of certain physical parameters. An air shock wave is a sudden variation of the pressure followed by a return to normal preceded by a phase of depression.

From : Militaty Ballistics  GM Moss, DW Leemings, CL Farrar 
Characteristics of a shock wave.
In an air environment, a pressure wave must meet a certain number of criteria to qualify as a shock wave :
1  The relative speed of the wave increases with the amplitude of the shock ;
2  The speed of sound behind the shock wave is always greater than the speed of the wave ;
3  A shock wave is always subsonic with respect to the medium behind it ;
4  The shock is always supersonic compared to the environment it meets ;
5  A shock wave is not an acceleration wave (which propagates at the speed of sound).
We will particularly note characteristic 4, which tells us that for a bullet to create a shock wave, it must have a supersonic speed in the medium in which it moves. The speed of sound in biological tissue is slightly higher than 1500 m/s. This is much higher than the velocity of high velocity bullets such as the .223 Remington (5.56 x 45mm). In the field of lesion ballistics, even these very fast bullets cannot create a shock wave in organic tissue. Therefore, one cannot attribute the observed lesions or part of them to the effect of a shock wave.
The image below represents the umbroscopy of a 7.62 mm bullet moving at a speed of 1.3 Mach.
V  TERMINAL BALLISTICS
Terminal ballistics corresponds to the interaction phase of the bullet with the target. Targets being of multiple natures, there are many specialties in terminal ballistics.
For small arms, the target is generally the combatant and the corresponding terminal ballistics is lesion ballistics.
In the following, we present some elements of wound ballistics.
V1  WOUND BALLISTICS
Wound ballistics is the study of the interaction between bullet and living tissue. The systematic analysis of the lesions has allowed the identification of a lesion pattern: the lesion profile.
• The concept of wound profile
Introduced by M. FACKLER and J. BRETAU, the wound profile, illustrated below, describes the interaction between the bullet and the target.
A : Proximal path (Neck)
B : Tumbling or mushrooming area
C : Distal path
• The three main modes of action of a bullet
Overall, when a bullet interacts with a target, it can exhibit three kinds of behavior :
1  Tilting or overturning ;
2  Expansion ;
3  Fragmentation ;
Below is a drawing showing these three modes of action.
The graphs below are the results of experiments. They illustrate the correlation between the attitude of the bullet in target, its deceleration and the importance of the woundss.
• Different bullets, different effects
The bullets are given different shapes and structures depending on the desired effect on the target.
• Tests  Reference materials
Experimentally, one method for studying the lesion potential of bullets consists of firing on reference materials (gelatin, ballistic gels), somewhat improperly called "simulants".
Analyses of feedback from the field concerning the lesionary effects of bullets, associated with the study of their behaviour on "simulants", make it possible to generate transfer matrices allowing the extrapolation of observations on "simulants" to the lesionary effects likely to be observed on living tissue.
Below are two examples of shooting on gelatin blocks.
READ MORE ABOUT WOUND BALLISTICS
