Analytical solutions of simplified equations оf motion in ballistic problems of a material point

Abstract

The paper substantiates the application of Lambert's special function for solving ballistic problems of a material point taking into account the resistance of gaseous (air) medium. Analytical solutions of simplified equations of motion for gentle and steep (relative to the horizon) trajectories are given. A comparative analysis of the results obtained using the Lambert function with the classical results obtained by Didion is carried out for the calculation of a hollow (hovering) flight trajectory. For the case of steep (relative to the horizon) flight trajectories of a material point, analytical solutions of the problem of external ballistics in quadrature have been obtained for the first time. The results obtained in this paper can be used to refine and improve the existing engineering methods for calculating the trajectories of motion and its main characteristics in the problems of external ballistics of a material point, when the nonlinear (proportional to the square of the velocity of motion) resistance of the medium is taken into account, in the problems of modern construction production, describing the processes of shotcrete concrete mixtures, as well as for the identification of the main parameters of motion, in particular, the coefficient of sailing.