Modelagem matemática do movimento bidimensional de projéteis sob a resistência do ar

Abstract

The study of projectile launches, both in secondary and higher education, is structured for the case of structured for the case of a vacuum, using concepts from kinematics. differences in relation to real observations, which occur under the effect of air resistance. In this way, the aim of this research was to develop a physical-mathematical study, using Dynamics, and to structure a model that more accurately describes the main characteristics of two-dimensional projectile movement under air resistance, using the equations of horizontal and vertical movement, as well as the equations for ascent time, maximum height, flight time, range and trajectory. To do this, we used the methods for solving differential equations, Taylor's series expansion, numerical calculation and simulations using Excel and Geogebra. For these simulations, three different spherical bodies were chosen (marbles, billiards and footballs), making comparative with vacuum launches and discussing the results for each of the bodies individually. bodies individually. Given the mathematical difficulties imposed by time of flight, three methods have been developed to obtain it. Two of them are Taylor series expansions, resulting in second (good approximation) and fourth (excellent approximation) degree equations. (good approximation) and fourth (excellent approximation) degree equations. The third, and simplest, consists only of multiplying the ascent time by the factor 2.2, producing an result for the range. Finally, a projectile launcher was built using low-cost materials to validate and compare the theoretical model. The main results show that the values determined by the mathematical model were satisfactorily close to those measured in the experiment with the launcher. Therefore, the equations established in this work are therefore applicable to problems involving problems involving two-dimensional launches of spherical projectiles under air resistance, without air resistance, without the occurrence of winds. Keywords: Projectile Launching; Dynamics; Air Resistance.