Effect of Aerodynamic Drag Forces

Second draft: 17 February 2000.

Fine tuning: 18 February 2000.

This document was originally writtin in Latex as a single file. A modified version was processed through latex2html, then the resulting HTML was edited manually to produce the more familiar linkage into sections and embedded graphics that you find here and in the following sections.

0. Overview

A Frequently Asked Question in the sci.physics newsgroup concerns how to include the effect of aerodynamic drag forces in calculations of projectile motion. As we see in Section 1, all realistic drag forces are sufficiently complicated that only a numerical solution is feasible for the general problem. I have done such an analysis for a specific set of questions that arise in baseball in a separate study. Nonetheless, the equations can be solved analytically if some plausible assumptions about the v dependence of the drag force are made. Further, those analytic solutions can be used to estimate the drag force on some arbitrary object based on a fit to free-fall data as the first step in a more detailed study of the motion of that object.

Note that the way this new version of latex2html works, equations are made extra large and then reduced to half-size to fit in the text, which can lead to invisible equal signs and other defects. Use the right mouse button to "view image" to see the equation in its original size to be sure you have it right before using it. Also note that I use italics in the text to minimize graphics for simple expressions. This makes a v (vee) for velocity look a lot like a $\nu$ (greek nu) for kinematic viscosity. The right meaning should be clear from context and whether an image was used within the text. Also note that links to the references take you to this page, so you should use the "Back" button on your browser to return after checking one of those (and ditto if you use a link to check an equation in another section than the one you are reading).

The outline of this document is as follows:

  1. Introduction to aerodynamic drag forces

  2. General solution for simplified (polynomial) forces

  3. Analytic solution for the K1 v case

  4. Analytic solution for the K2 v2 case

  5. Sample results

  6. Fitting data to get CD equation


Bibliography

1
Russell L. Donnelly in Section 12 of "A Physicist's Desk Reference: The Second Edition of Physics Vade Mecum", ed. Herbert L. Anderson (AIP, New York, 1989), pages 197ff.

2
"The Physics of Baseball" by Robert Adair.


Jim Carr
2000-02-17