External ballistics howitzer projectile
Authors: Velychko Lev , Petruchenko Oksana , Tereshchuk Oksana , Nanivskyi Roman 
Published: May 2021

In this scientific work, the team of authors presents a mathematical model for studying the dynamics of the motion of a projectile in the air, fired from cannon.
One of the main problems of external ballistics is to determine the magnitude of the force of the air resistance to the movement of the projectile. Usually in studies, a discrete relationship between the magnitude of the force of resistance and projectile velocity has been established. However, to improve the accuracy of firing, it is necessary to determine the functional dependence of air resistance on projectile velocity, deterministic and non-deterministic factors. The authors, when processing the results of landfill studies, which are presented in the tables of firing, found that the magnitude of the force of air resistance to the movement of the projectile depends not only on its speed but also on acceleration Based on this, the functional dependence of the force of air resistance is described separately during the movement of the projectile with the following velocities: supersonic (stage I); subsonic - with negative acceleration (stage II); subsonic with positive acceleration (stage III).
To determine the coefficients of functional dependences, it is proposed to use inverse dynamics problems. Boundary conditions were considered - the full horizontal range of the projectile, depending on the specific angle of impact, obtained from the results of landfill research and given in the firing tables.
Under the condition of a certain functional dependence of the force of counter-air resistance, taking into account the weight of the projectile and the Carioles’ force, as a result of this work is obtained the system of differential equations, which describes the motion of the projectile in air. The initial conditions for the first stage were taken the initial velocity of the projectile and zero (original) coordinates; for the second stage - the value of the kinematic parameters of the projectile at a time when its speed became equal to the speed of sound in the air; for the third stage - the value of the kinematic parameters of the projectile at the time when its velocity began to increase.
By solving the system of differential equations, using the appropriate software, can be determined the impact of projectile charge and air temperatures, atmospheric pressure, changes in projectile mass and its initial velocity on the kinematic parameters of projectile motion. In addition, it allows you to automate the process of determining the aiming angle (it is better to ask the gunners the correctness of this concept) depending on the firing range, taking into account the above factors.
Also, in the work on the basis of the method proposed by the authors, the is carried out comparison of the kinematic parameters of the projectile with the results given in the firing tables. They indicate minor differences when shooting at short distances, but when shooting at long distances - these differences increase, as the results in the tables of shootings are quite approximate.