GENERAL SOLUTION OF TWO-DIMENSIONAL PROJECTILE MOTION WITH AIR RESISTANCE

In this study, two-dimensional projectile motion is considered under the effect of a general power law model of air resistance. Classically, a projectile is treated as a point mass with mass m moving in a uniform gravitational field. The projectile is launched from the ground with an angle α to horizon. the drag force is assumed to be proportional to the speed raised to the power n. The analysis of the problem is performed using Cartesian coordinates. A general exact parametrical solution (with respect to the angle of motion) is derived for any power n, following simple steps: 1) find the speed in the direction of the axis x (horizontal – no gravity) 2) find the vertical component of the speed 3) find the time 4) find the horizontal position of the projectile and finally,5) find the vertical position of the projectile. Steps 1) and 2) give explicit closed form equations and the rest are given by exact integrals which can be solved numerically. In this study spreadsheet calculation are performed using trapezoidal rule of integration. The cases of motion in avacuum and linear drag law are used to check the accuracy of the numerical calculations.The importance of the proposed study is three-fold: a) The method of the derived solution is new, and couldnt be found elsewhere b) The derived equations make it possible to use spreadsheets for presenting the subject (no programming is required), and thus, serve as a tool to enhance teaching c) The derived equations are general for any power n thus, the same procedure could be used to find the position of the projectile at any time.