Impact of the Drag Force and the Magnus Effect on the Trajectory of a Baseball

Abstract

How to cite this paper: Sarafian, H. (2015) Impact of the Drag Force and the Magnus Effect on the Trajectory of a Baseball. Abstract We consider the impact of drag force and the Magnus effect on the motion of a baseball. Quantitatively we show how the speed-dependent drag coefficient alters the trajectory of the ball. For the Magnus effect we envision a scenario where the rotation of the ball confines the Magnus force to the vertical plane; gravity, drag force and the Magnus force make a trio-planar system. We investigate the interplay of these forces on the trajectories. Keywords Drag Force, Magnus Effect, Spinning Ball, Baseball, Nonlinear Physics, Mathematica 1. Motivation and Introduction In introductory physics, engineering and math courses, undergraduate students are traditionally introduced to the concept of projectile motion such that a projectile is thrown at an angle in a vertical plane in a vacuum. The motion in the vacuum is analyzed because in the absence of air, utilizing the Newton's second law yields to a set of two independent second-order linear ODEs. The trivial solution of these equations provides information about the kinematics of the projectile, such as the trajectory, range, time of flight, etc. [1]. If one applies the learned concepts to a real-life situation such as a pitched and/or a batted baseball, one disappointingly would be surprised realizing the gross discrepancies of the two scenarios. For instance, a 45˚batted45˚batted baseball does not maximize the range, and the trajectory of a baseball is not a symmetric parabola. The motivation of this article stems from the former question, namely "At what angle above the horizontal a baseball should be batted making the range a maximum?" To answer this question, we analyze the problem systematically. In the course of our analysis, we stumbled upon a few interesting related issues. To the author's amazement, the analysis led also to uncharted territories unveiling features of a flying baseball that to date have not been reported in scientific literature. In this article we confirm the previously reported quantitative results