Abstract. Various models of a projectile in a resisting medium are used. Some are very simple, like the "point mass trajectory model", others, like the "rigid body trajectory model", are complex and hard to use, especially in Fire Control Systems due to the fact of numeric complexity and an excess of less important corrections. There exist intermediate ones -e.g. the "modified point mass trajectory model", which unfortunately is given by an implicitly defined differential equation as Sec. 1 discusses. The main objective of this paper is to present a way to reformulate the model obtaining an easy to solve explicit system having a reasonable complexity yet not being parameter-overloaded. The final form of the M-model, after being carefully derived in Sec. 2, is presented in Subsec. 2.5.Key words: ballistics, equations of motion, projectile path, modified point mass trajectory model, MPMTM, projectile deflection.
Preliminaries: physical assumptionsand common models 1.1. Brief discussion of different ballistic models and their physical assumptions. We decided to start our work by discussing physical basics of modelling a projectile's trajectory in a resisting medium. Most of the statements in this section are meant as a preliminary and are made rigorous in progress of this paper. We discuss the general properties of three typical models:• Commonly by the "point mass trajectory model" one understands a very simple physical model of the trajectory of a projectile in which the only acting forces are external forces, e.g. gravity, and the head-on drag force. Such a model describes a movement that is planar and does not explain such phenomena as the drift caused by gyroscopic precession and thus induced lift forces.On the other hand, one can form a highly sophisticated "rigid body trajectory model" of a ballistic projectile, which treats the projectile as a six-degree-of-freedom physical object with axial symmetry and non-zero moments of inertia I x and I y , being respectively the moment of inertia along the axis of the projectile and the moment of inertia in the plane perpendicular to the projectile's axis. This model has the capabilities of explaining most physical phenomena happening along the flight trajectory but is unfortunately quite cumbersome. In addition, such "rigid body model" requires as input a huge number of various coefficients which cannot be fitted to poor quality experimental data, or even worse -their fitting might be considered highly inaccurate and ambiguous. To derive equations of motion for aerial objects treated as rigid bodies mostly Newtonian approach is used, i.e. forces, momenta or momentum and angular momentum conservation laws. Examples could be found in [7][8][9]. Sometimes more involved theoretical mechanics is used, e.g. Lagrangian formulation, see [10], or Boltzmann-Hamel equations like in [11]. In this paper, as it is described, we use a four degree of freedom model and to keep things simple -restrict ourselves to forces and momenta.To find a moderately complex solution but still expl...
