One dimensional motion with linear plus quadratic drag

Live Script providing symbolic and numerical calculations of 1-dimensional motion with linear plus quadratic drag

A simple model of the effect of fluid on an object in motion is a drag force linear in velocity for low velocity and proportional to the square of speed at high velocity. This Live Script derives and explores analytic solutions to Newton's equation for ballistic motion in one dimension of a mass subject to a constant gravitational force and a combined linear plus quadratic drag force:

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The analytic solutions depend on the direction of initial velocity relative to the direction of gravity and are used piecewise to describe a general initially upward-going ballistic motion followed by a downward motion. A maximal time to reach the peak height, independent of initial speed, is derived as well as expressions for the terminal velocity. The symbolic results are compared to numerical calculations and exported to functions for general use.

This Live Script may interest students and instructors in physics and other disciplines. The results might be used to compare experimental free fall data to the drag model using a nonlinear fitting technique. The calculations illustrate a variety of symbolic and numerical integration techniques.