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Projectile Motion with Aerodynamic Drag: The Cubic Law

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2015 ASEE Annual Conference & Exposition


Seattle, Washington

Publication Date

June 14, 2015

Start Date

June 14, 2015

End Date

June 17, 2015





Conference Session

Engineering Physics & Physics Division Technical Session 4

Tagged Division

Engineering Physics & Physics

Tagged Topic


Page Count


Page Numbers

26.1273.1 - 26.1273.31



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Paper Authors


Jeffrey C. Hayen Oregon Institute of Technology

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Jeffrey Hayen joined the faculty in the MMET Department at the Oregon Institute of Technology (OIT) in 2011. Before arriving at OIT, Jeffrey served as a Professor of Engineering, Mathematics, and Physics at Southwestern Oregon Community College for 16 years. Prior to that experience, he worked in the aerospace industry as a thermodynamicist and propellant analyst for high-performance upper-stage rockets at the Space Systems Division of the General Dynamics Corporation. He also has conducted research concerning structural dynamics and control for the Kajima Corporation of Japan, and he currently provides technical analyses and performs computational simulations for the United Launch Alliance in Denver. Jeffrey earned his B.S. and M.S. degrees in Mechanical Engineering from San Diego State University, and his Ph.D. degree in Applied Mechanics and Physics from the California Institute of Technology.

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Projectile Motion with Aerodynamic Drag: The Cubic LawA classic problem covered in engineering mechanics and physics courses is the determination ofthe trajectory for projectile motion. The version of the problem traditionally introduced does notinvolve any resistance effect due to the medium through which the projectile travels. However,in terrestrial applications, the atmosphere present inherently produces an aerodynamic drag forcethat noticeably alters the trajectory in comparison to the trajectory that would occur in a vacuum.The primary reason for neglecting the drag force is the mathematical complications that arise inthe governing equations of motion if a drag force effect is included.In realistic applications, it can be shown that the drag force on the projectile can be appropriatelymodeled as being proportional to projectile-speed squared. An exact solution to the equations ofmotion for this model can be obtained, but it involves quadratures which cannot be analyticallyevaluated in terms of standard functions. An approximate solution that is remarkably accurate ispresented in this article. This solution is based upon the so-called cubic law, which is motivatedby certain properties of the exact solution. It involves a third-degree polynomial equation whichis utilized to generate the projectile trajectory. Because of the relative simplicity of this familiaralgebraic equation, estimates can be readily obtained for other quantities of interest, such as themaximum projectile range and optimal projection angle. Although they are approximate, theseestimates provide insight to students about the essential differences in the results for the realisticand idealized versions of this classic problem.An intriguing characteristic of the approximate solution for projectile motion with aerodynamicdrag included is that it has the same form as the exact solution for projectile motion in a vacuum(a well-known second-degree polynomial equation, whose graph is an inverted parabola), exceptfor the addition of a single cubic term.The author has utilized the approximate solution to create an engaging programming exercise forstudents enrolled in an engineering computation course. In this course, the students are tasked toconstruct an EXCEL® spreadsheet with charts (graphs) to display the trajectories for a projectileunder a variety of conditions. It is anticipated that this spreadsheet will be briefly demonstratedduring the formal presentation of this article.

Hayen, J. C. (2015, June), Projectile Motion with Aerodynamic Drag: The Cubic Law Paper presented at 2015 ASEE Annual Conference & Exposition, Seattle, Washington. 10.18260/p.24610

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