Approximate Analytical Description of the Projectile Motion with a Quadratic Drag Force

Abstract

In this paper, the problem of the motion of a projectile thrown at an angle to the horizon is studied. With zero air drag force, the analytic solution is well known. The trajectory of the projectile is a parabola. In situations of practical interest, such as throwing a ball with the occurrence of the impact of the medium the quadratic resistance law is usually used. In that case the problem probably does not have an exact analytic solution and therefore in most scientific publications it is solved numerically. Analytic approaches to the solution of the problem are not sufficiently advanced. Meanwhile, analytical solutions are very convenient for a straightforward adaptation to applied problems and are especially useful for a qualitative analysis. That is why the description of the projectile motion with a simple approximate analytical formula under the quadratic air resistance represents great methodological interest. Lately these formulas have been obtained. These formulas allow us to obtain a complete analytical description of the problem. This description includes analytical formulas for determining the basic eight parameters of projectile motion. Analytical formulas have been derived for the six basic functional dependences of the problem, including the trajectory equation in Cartesian coordinates. Also this description includes the determination of the optimum throwing angle and maximum range of the motion. In the absence of air resistance, all these relations turn into well-known formulas of the theory of the parabolic motion of the projectile. The proposed analytical solution differs from other solutions by simplicity of formulas, ease of use and high accuracy (relative error is about 1-2 %). The motion of a baseball is presented as an example. The proposed formulas make it possible to carry out an analytical investigation of the motion of a projectile in a medium with resistance in the way it is done in the case of no drag.