Abstract
This article centers around the problem of maximizing the horizontal range of a projectile that is launched from atop a tower and is subject only to gravity and air resistance quadratic in speed. Here the surface to which the projectile is launched is represented by a convex impact function, while the projectile motion is described by a classical approximation model for flight curves that is widely considered acceptable for quadratic drag and launch angles up to moderate size. In this setting, the optimal range is given by the point where the impact function intersects the enveloping function induced by the family of flight paths. In the special case of a linear impact function, manageable explicit formulas for the range function, the maximal range, and the corresponding optimal launch angle are provided in terms of the Lambert W function. The article concludes with a solution to Tartaglia’s inverse problem in this context.
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References
Belgacem, C.H.: Range and flight time of quadratic resisted projectile motion using the Lambert \(W\) function. Eur. J. Phys. 35, 055025–055032 (2014)
Bruckner, A.M., Bruckner, J.B., Thomson, B.S.: Real Analysis. Prentice-Hall, Upper Saddle River (1997)
de Mestre, N.: The Mathematics of Projectiles in Sport. Cambridge University Press, Cambridge (1990)
Groetsch, C.W.: Tartaglia’s inverse problem in a resistive medium. Am. Math. Mon. 103, 546–551 (1996)
Groetsch, C.W.: Inverse Problems. Mathematical Association of America, Washington, D.C. (1999)
Groetsch, C.W.: Geometrical aspects of an optimal trajectory. Pi Mu Epsilon J. 11, 487–490 (2003)
Hackborn, W.W.: Motion through air: what a drag. Can. Appl. Math. Q. 14, 285–298 (2006)
Hackborn, W.W.: Projectile motion: resistance is fertile. Am. Math. Mon. 115, 813–819 (2008)
Halley, E.: A proposition of general use in the art of gunnery, shewing the rule of laying a mortar to pass, in order to strike any object above or below the horizon. Philos. Trans. R. Soc. 19, 68–72 (1695)
Kantrowitz, R., Neumann, M.M.: Optimal angles for launching projectiles: Lagrange vs CAS. Can. Appl. Math. Q. 16, 279–299 (2008)
Kantrowitz, R., Neumann, M.M.: Let’s do launch: more musings on projectile motion. Pi Mu Epsilon J. 13, 219–228 (2011)
Kantrowitz, R., Neumann, M.M.: Some real analysis behind optimization of projectile motion. Mediterr. J. Math. 11, 1081–1097 (2014)
Kantrowitz, R., Neumann, M.M.: Optimization of projectile motion under linear air resistance. Rend. Circ. Mat. Palermo 64, 365–382 (2015)
Lamb, H.: Dynamics. Cambridge University Press, Cambridge (1961)
Packel, E.W., Yuen, D.S.: Projectile motion with resistance and the Lambert \(W\) function. Coll. Math. J. 35, 337–350 (2004)
Parker, G.W.: Projectile motion with air resistance quadratic in the speed. Am. J. Phys. 45, 606–610 (1977)
Stromberg, K.R.: Introduction to Classical Real Analysis. Wadsworth, Belmont (1981)
Warburton, R.D.H., Wang, J.: Analysis of asymptotic projectile motion with air resistance using the Lambert \(W\) function. Am. J. Phys. 72, 1404–1407 (2004)
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Kantrowitz, R., Neumann, M.M. Optimization of Projectile Motion Under Air Resistance Quadratic in Speed. Mediterr. J. Math. 14, 9 (2017). https://doi.org/10.1007/s00009-016-0815-4
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DOI: https://doi.org/10.1007/s00009-016-0815-4