Research of Influence Ballistic Characteristics of Weapons on the Shooting Efficiency Taking Into Account the Safety of the Small Arm Use

The object of research is the process of performing a fire mission by employees of the security forces in the presence of unauthorized persons in the direction of fire. The paper investigates the influence of the ballistic characteristics of weapons on the effectiveness of performing fire missions by security forces, taking into account the security of the use of weapons. The purpose of the use of weapons by security forces is usually the cessation of the offender or its detention. In this case, the deaths or injuries of a hostage or an unauthorized person as a result of the use of weapons should qualify as failure to fulfill the assigned fire mission. An analysis of the tactics of actions and features of the use of weapons by security forces indicates that the main reason for the danger of the use of small arms for unauthorized persons is the excessive distance of the bullet’s flight, at which it maintains lethal action. At the same time, with this the distance of the lethal action of a bullet is one of the factors that can be corrected. It depends on a combination of the ballistic characteristics of the weapon, namely the muzzle speed and the ballistic coefficient of the bullet. The analysis of known studies shows that the existing scientific and methodological apparatus for determining the effectiveness of firing does not allow to determine the effectiveness of the execution of the fire mission, taking into account the safety of the use of small arms due to the imperfection of the corresponding models. The model of the process of performing the fire mission has been improved by taking into account the influence of the ballistic characteristics of the weapon on the probability of hitting unauthorized persons, which makes it possible to assess the safety of the use of weapons. As a result of the study of the influence of ballistic characteristics of weapons on firing efficiency, taking into account the safety of the use of weapons, it was found that there is a need to minimize the difference between the aiming range of the weapon and the range at which the bullet maintains lethal action. Excessive energy of a bullet increases the likelihood of hitting an outsider due to an increase in the area of the danger zone of its destruction as a result of through penetration of the target. The probability of performing a fire mission with a limitation on the safety of using weapons is positively affected by increasing the stability of the muzzle speed of a bullet.


Introduction
In the design and manufacture of small arms for the defense forces, considerable attention is paid to its ballistic characteristics. The most important of them are the muzzle speed of a bullet and the dynamics of a drop in speed on the trajectory, which is due to the ballistic coefficient of the ball. The value of ballistic characteristics is determined by such tactical characteristics of the weapon as the aiming range, the range of the direct shot and the range of the lethal action of the weapon, as well as the piercing, stopping and lethal action of the bullet. The higher the specified charac teristics, the more chances to hit the target, that is, the shooting efficiency increases. Along with this, the likelihood of defeating any other persons and objects in the direc tion of fire increases, but this is not a problem, since there should not be civilians or friendly forces in enemy positions.
The purpose of the use of weapons by the security for ces (SF) in the vast majority of cases is the cessation of the offender or its detention. Moreover, neglect of the life or health of a member of the society whom these forces protect from the offender is not acceptable. Cases of the death or injury of a hostage or an unauthorized person as a result of the use of weapons during the performance of military service missions of the SF should qualify as failure to fulfill the assigned fire mission (FM). However, in the practical use of weapons by security officers, there is a possibility of a bullet falling into an unauthorized person and its defeat.
Thus, for SF, the need to simultaneously ensure a suf ficient probability of hitting a target and minimizing the likelihood of hitting unauthorized persons. At the same time, increasing the efficiency of shooting due to the above tactical characteristics of weapons is not possible. So, the study of the influence of ballistic characteristics of weapons ISSN 2664-9969 on the effectiveness of shooting, taking into account the security of the use of small arms, is relevant.

The object of research and its technological audit
The object of research is the process of performing a fire mission by SF employees in the presence of unauthorized persons in the fire direction.
The components of the system in which this process takes place include: -subject of the use of weapons is an employee of the security forces; -object of the use of weapons -the enemy (target); -sample of small arms; -unauthorized persons who are in the direction of fire and are located at a distance of no less than the target and less than the distance of the lethal action of the bullet. The aim of the process is to defeat the object of use of weapons. The process of performing a fire mission is characterized by: -distance to the target; -location of unauthorized persons in space; -ballistic characteristics of weapons; -accuracy of pointing the weapon at the target; -technical accuracy of the shooter; -probability of hitting the target; -evidence of nondefeat by unauthorized persons; -time to complete the fire mission. Limitations on the process flow: -minimum probability of hitting a target; -minimum safety of the use of weapons; -maximum time to hit the target. In the process of performing a fire mission due to vari ous adverse conditions, there is a likelihood of an outsider being struck, which is considered unacceptable. Adverse conditions include: -low preparedness of the shooter; -large distance to the target; -significant target mobility; -large number of unauthorized persons; -the proximity of unauthorized persons to the target; -significant time constraints on the performance of the fire mission; -low technical accuracy of weapons; -large distance of the lethal action of a bullet and the like. An analysis of the tactics of action and the specifics of the use of weapons by security forces indicates that the main reason for the danger of using small arms of the SF is the excessive distance of the bullet's flight, at which it maintains the lethal effect of Х LE . When shooting at a target, there is always a chance of a miss. At the same time, when using a weapon, a security officer sees the space well to the target. But it does not concentrate on the situation behind the target and does not have the ability to control it in the direction of fire at a depth of the distance of the bullet's flight, at which it maintains a lethal effect. Therefore, there is the possibility of fall ing into a stranger and its defeat. At the same time, the slaughter distance of the ball is one of the factors that can be corrected. It depends on the ballistic characteristics of the weapon, in particular on the muzzle speed and ballistic coefficient of the bullet. With certain combinations of bal listic characteristics of weapons, the Х LE is too large, which increases the likelihood of hitting unauthorized persons and reduces the likelihood of a fire mission as a whole.
Thus, the characteristic disadvantages of the object of research are the irrational ballistic characteristics of the weapon and the resulting unreasonably high probability of hitting an outsider.
To increase the security of the use of weapons by security forces, it is necessary to know how the distance of the lethal action of a bullet affects the likelihood of a fire mission with restrictions on the security of weapon use (SWU).

The aim and objectives of research
The aim of research is to study the influence of bal listic characteristics of weapons on firing efficiency, taking into account the safety of small arms.
To achieve this aim, it is necessary to complete the following objectives: 1. Improve existing models of firing efficiency in order to take into account the safety of the use of weapons.
2. Determine the effectiveness of the execution of the fire mission by law enforcement forces depending on the ballistic characteristics of weapons in peacetime.

Research of existing solutions of the problem
Enough work has been devoted to determining the ef fectiveness of shooting from small arms. So, in [1], applied aspects of probability theory are considered, in particular, general methods for determining the probabilities of hitting a target and hitting a target. The work [2] is devoted to the methodology of firing efficiency. In particular, it considers a generalized performance indicator, the conditional law of hitting a target, assessing factors affecting the accuracy of hitting a target, models for assessing the probability of hitting a target, taking into account counteraction to the like. These works can be used to develop models for performing a fire mission taking into account SWU, but these issues are not directly addressed in the work.
The authors of [3] highlight a wide range of ques tions from the basic information of probability theory to an experimental study of firing efficiency. The work also examined the laws of target destruction, criteria for firing efficiency and combat effectiveness, taking into account the enemy's fire counteraction, and the like. However, all the material is devoted to the combat use of weapons without taking into account SWU.
In [4], the authors focus on the theory of fire duel ing, in particular, on the effectiveness of hiding and air dueling, as well as on methods for synthesizing fire con trol strategies. At the same time, aspects of the use of weapons by security forces in the work are not covered.
In studies [5], tasks and general principles for assess ing combat effectiveness, combat effectiveness indicators, methods for determining the probability of defeating a small and a large target are considered. The paper also provides estimates of the economic effectiveness of weapons and statistical methods for assessing the combat effective ness of weapons. But the above mathematical apparatus does not allow to assess the safety of using weapons for unauthorized persons.

ISSN 2664-9969
The work [6,7] is devoted to questions of theory and practice of choosing the optimal characteristics and parameters of barrel complexes according to the relevant performance criteria. However, the use of small arms is not addressed in these studies.
In [8], a method for calculating the penetration of sniper shells into specific targets is considered, but the issue of ensuring the safety of unauthorized persons is not considered.
A common feature of the above works is that they consider the use of weapons by the defense forces and do not take into account the presence of objects whose destruction is not desirable.
The work [9] shows a model for the response of specific targets to impacts at high speed, providing the dominant contribution of the friction shift. It reveals the protective capabilities of the shelter of military personnel during hos tilities. It is not talk about ensuring the safety of civilians.
The study [10] shows the limited suitability of exist ing indicators and criteria for the effectiveness of firing for evaluating the effectiveness of specific fire tasks by security forces. A new indicator of the reliability of the performance of the fire mission by the security forces is developed, taking into account the probability of an un authorized person's defeat. Along with this, the work does not address the dependence of firing efficiency indicators on the ballistic characteristics of weapons.
Thus, the results of the analysis allow to conclude that the existing scientific and methodological apparatus for determining the effectiveness of firing does not allow to determine the effectiveness of the execution of the fire mission, taking into account the SWU.

Methods of research
The likelihood of an unauthorized person being affec ted is influenced by several factors.
Target characteristics: Given that the likelihood of a slip indirectly increases the likelihood of hitting an unauthorized person, we can conclude that the probability of hitting an unauthorized person is positively affected by: -the number of unauthorized persons; -dimensions of the frontal projections of unauthori zed persons; -distance to the target; -hits RMS in the target plane in height and lateral direction (up to a certain value); -HM deviation from the center of the target (to a certain value). In this case, negatively affect the likelihood of hitting an unauthorized person: -dimensions of the frontal projection of the target; -distance to unauthorized persons; -deviation of the centers of unauthorized persons from the center of the target, starting from the values equal to the sums of half of the corresponding dimen sions of the target and the unauthorized person.
To study the influence of the distance at which the bullet maintains a lethal effect on the SWU, a program is developed in the Mathcad environment.

Research results
The initial data for the calculations are the values corresponding to the factors that influence the probability of an unauthorized person hitting: Y С , Z С , X С , y HM , z HM , σ y , σ z , Y і , Z і , X і , y і and z і .
Given that the RMS in the target plane depends on the distance to it, the scattering characteristics at a stan dard distance of 25 m for the gun are taken as the basis. For other distances that correspond to the position in space of the target and unauthorized persons, the RMS is calculated using the formulas: Y HM value has to be adjusted depending on the distance to the target. This is due to the movement of the ball along the coordinate in under the action of gravity. So, for example, a change in the ball's position in height at an aiming distance (200 m) for a 9mm pistol is up to 1.5 m, and for a 5.45mm automatic machine AKS74U at a distance of 500 m -0.95 m not to be ignored.
To correct the value of y HM , equation [11] is used, which allows one to calculate the air drag force of PE motion: where R xa -air drag force; C xa -aerodynamic drag coefficient; ρ -density of the atmosphere; v -speed of PE movement; S m -the area of the PE midsection, as well as the accelera tion of the drag force І R : where m -PE mass.
Electronic copy available at: https://ssrn.com/abstract=3679753 ISSN 2664-9969 Based on the wellknown dependence connecting the path of the body with its initial speed, acceleration and time of movement [12] let's obtain: From (8) let's find the time of PE movement in the selected section of the path S: and after substituting (8) in (7) let's obtain the expression for calculating the magnitude of the PE displacement along the coordinate у: In this case, the trajectory should be divided by distance into sufficiently small sections so that the PE speed at the beginning of the section does not differ significantly from the average value in this section. Calculations show that for the studied cases, the length of the plot of 0.05 m provides a quite acceptable relative error in determining the speed, and therefore the time -0.014-0.023 %. This is confirmed by the high convergence of the calculation results with the operational documentation data for weapons samples for which such data are given [13][14][15].
The calculation of the probability of hitting the target Р С and by unauthorized persons Р UP is carried out according to the formulas [3]: . d (12) The probability of hitting the target W С and the unautho rized person W UP і can be calculated by the expressions [10]: The probability of performing a fire mission by an SF employee, taking into account SWU, is calculated accord ing to the formula [10]: The indicated formula follows from (13) under the con dition that for hitting an outsider one bullet hit is enough and the energy characteristics of the ball are sufficient for hitting. That is, the case of the location of the target and unauthorized persons at distances not exceeding the range that ensures the lethal effect of the bullet is considered.
But in practice it can happen differently, so it is neces sary to consider cases where the action of the bullet on the target is excessive from the point of view of safety, insufficient to hit an outsider, and also uncertain due to the dispersion of the muzzle speed of the bullet. In this case, there are unfavorable probabilities of excessive Р exc and indefinite Р ind of the bullet on the target. Based on this, the probability of defeat and the first unauthorized person is equal to: and the probability of defeating at least one unauthorized person is equal to the sum of the probabilities W UP i : where m -the number of unauthorized persons. Thus, let's obtain: The probabilities Р ind and Р exc are calculated in accor dance with (8) and (9), and the ranges ΔХ ind according to the formula [16]: ln ln . min m in (19) The boundaries of the X min and X mах ranges are deter mined by the formulas [17]: where V m -PE muzzle speed; С -PE ballistic coefficient of PE; X min -distance at which the PE has a minimum energy that provides a lethal effect; X mах -distance at which the PE has maximum energy can be considered safe; V Х min -PE speed at a distance X min ; V Х max -PE speed at a distance X mах .
The corresponding values of V Хmin and V Хmax are calcu lated by the formulas [17]: The minimum and maximum values of the PE energy characteristics are set depending on the nature of the target and the degree of its protection. An analysis of literary ISSN 2664-9969 sources shows that the minimum energy that provides a lethal effect is approximately 20 J, and the maximum energy that can be considered safe is about 10 J [18].
The ranges ΔХ exc is equal to X min . For cases of PE reflection from an obstacle, the value V m is replaced by the value of the PE speed after reflection, and the direction of PE flight is determined from the condition that the angles of its meeting with the obstacle and reflection from it are equal.
The set of expressions (8)- (12) and (18)-(23) consti tute a model of the process of performing a fire mission with restrictions on the safety of using weapons.
An analysis of the model shows that a real way to increase the likelihood of performing a fire mission with restrictions on the safety of using weapons is to reduce the Р exc value, in particular, at distances exceeding the aiming after PE reflection from the obstacle.
In order to obtain realistic values of the parameters that affect SWU, an analysis was made of cases of the use of weapons by security forces, the level of small arms training of security personnel and the characteristics of small arms.
It is established [19] that in the vast majority of ca ses (98 %) pistols are used.
The shooting was mainly carried out from unstable positions, namely from a standing position (60 %), in mo tion (4.4 %) and from a car (20.5 %). In three quarters of the cases, firing was conducted from one hand.
Of the stable positions for shooting (from the knee, lying down, from cover) only about 15 % of the shots were fired.
The distance to the target did not exceed 20 m in 85 % of cases.
In 89 % of cases, the time for shooting was limited, and in 30 % of cases the shooting was fired without aim ing (offhand).
In 71 % of cases of the use of weapons, visibility was limited.
The place of use of weapons is distributed as follows: on the street -80 %, in parks and forests -8 %, in the premises -12 %.
More than half of the use of weapons is carried out in adverse conditions: physical activity -13 %, psychological stress -29 %, ontarget shooting -10 %.
The statistics provided allow to draw conclusions about the conditions of use of the weapon, the corresponding values of the errors of pointing the weapon at the target and the quality of the shot.
So, in accordance with [20], the error of bringing a weapon to a normal battle is up to 0.05 m at a distance of 25 m.
The size of the median error (В v and В b ) of pointing the weapon at the target when shooting on the move from a short stop is two thousandths [21], which for a distance of 25 m is also 0.05 m.
When shooting on the go or with a moving error will be even higher. The time limit for shooting also increases the value of the pointing error. The analysis of the re sults of highspeed shooting from a distance of 25 m from cadets of various courses of the academy gives the total error of pointing the weapon at the target and bringing the weapon to normal combat у HM = z HM = 0.12-0.27 m.
Accuracy of shooting from small arms when brought to normal combat is considered acceptable provided that R100 = 0.15 m [13,14,20], that is, RME of hitting the target plane is σ = 0.025 m. But when performing fire tasks in traffic conditions and time limits the accuracy of shooting is significantly reduced for the studied groups of cadets σ у = σ z = 0.08-0.20 m.
The  Fig. 1 shows the placement of unauthorized persons (fi gures in blue), against which the target can be located (figure in red). The scheme is made according to the photograph and corresponds to the real location of citizens on the sta tion square in Kharkiv (Ukraine). Fig. 2 shows the dependences of the probabilities of hitting a target and performing a fire mission on the range of the lethal action. The triangular and rhomboid markers reflect the values of W SWU and W UP for the corresponding Х LE distances corresponding to the location of unauthorized persons in Fig. 1. The solid curve reflects the dependen cies for a similar case, but for uniform distribution of unauthorized persons in distance and lateral direction.
For other values of the number of unauthorized persons, the distance to the target, the distance of the PE lethal action, the accuracy of firing, etc. curves are obtained that are similar in nature to those given above (Fig. 2).
An analysis of the obtained dependencies shows that the increase in the probability of an unauthorized person being injured gradually decreases with an increase in Х LE . This is due to the linear dependence of the number of people in the danger zone from Х LE and the quadratic de pendence of the area of projections of unauthorized persons on the distance to them. So, the increase in the area of the frontal projections of unauthorized persons, taking into account their angular dimensions, lags behind the increase in the distance of the lethal action.
The dependence W SWU (Х LE ) follows from the depen dence W UP (Х LE ) taking into account expressions (13)- (15). With a decrease in Х LE , the probability of completing a fire mission increases progressively. For the given example, the increase in W SWU for every 50 m of reduction in the distance of slaughter action is given in Table 1.  In the case when the energy characteristics of PE ex ceed the values that are necessary for a reliable hit of the target of the above danger of hitting an outsider due to a miss by the target, the danger of hitting it with a bullet that pierced the target through and through can be added.
It should be noted that the ΔХ ind value significantly affects the stability of the muzzle speed of a bullet, follows from expression (19). The expansion of the ΔХ ind range negatively affects the efficiency of the fire mission. The value of V m depends on a number of factors, in particular, on the values of the charging parameters [23]. Therefore, to ensure the stability of the values of the charging pa rameters, considerable attention should be paid. Possible ways to increase the stability of the muzzle speed of the striking elements of kinetic weapons are given in [24].

SWOT analysis of research results
Strengths. Strengths of research ire the use of well tested classical approaches in the field of external ballis tics and firing efficiency, which ensures high reliability of forecasts. In practical terms, a significant positive effect is expected from the ability to reduce the likelihood of hitting unauthorized persons and increase the effectiveness of the performance of fire missions by security forces.
Weaknesses. Weaknesses include the difficulty of prac tical implementation of the requirement to minimize the difference between the aiming range of a weapon and the range at which the bullet maintains a lethal effect. This is due to the low stability of the muzzle speed of a bullet when using insufficiently highquality ammunition (long term storage, various manufacturers, etc.).
Opportunities. Studies of the opportunities of abrupt ly reducing the bullet speed at the aiming distance of the weapon seem to be promising, which will mini mize the difference between the aiming range of the weapon and the range at which the bullet preserves the lethal effect.
Threats. The practical implementation of the ideas is associated with the need to improve the quality of ammunition or develop new cartridges with an abrupt decrease in bullet speed, which requires additional ma terial costs.

Conclusions
1. The analysis of known studies shows that the exist ing scientific and methodological apparatus for determining the effectiveness of firing does not allow to determine the effectiveness of the execution of the fire mission, taking into account the safety of the use of small arms due to the imperfection of the corresponding models. Therefore, the model of the process of performing the fire mission is improved by taking into account the influence of the ballistic characteristics of the weapon on the likelihood of hitting unauthorized persons. This creates an opportunity to assess the safety of the use of weapons.
2. The influence of the ballistic characteristics of the weapon on the effectiveness of firing, taking into account the safety of the use of small arms, is investigated. It is established that the nature of the dependence of W SWU (Х LE ) indicates the need to minimize the difference between the aiming range of the weapon and the range at which the bullet maintains a lethal effect. Excessive energy of a bul let increases the likelihood of hitting an outsider due to an increase in the area of the danger zone of its destruc tion as a result of through penetration of the target. The probability of performing a fire mission with a limitation on the safety of using weapons is positively affected by increasing the stability of the muzzle speed of a bullet.