[math-fun] projectile with quadratic air drag solution
I found this on the web posted by "pelli" on "reddit". Brilliant! Fantastic! Here's a forward solution (found by reverse-engineering the answer): Consider a projectile moving in gravity with quadratic air resistance. The governing equations are u' = -a * u * sqrt( u^2 + v^2 ) v' = -a * v * sqrt( u^2 + v^2 ) - g where a is the coefficient of air resistance defined by |F| = ma|v|2 . Cross-multiply and rearrange to find a * sqrt( u^2 + v^2 ) * (u*v'-v*u') = g*u' Substitute v = s*u and separate variables: a * sqrt( 1 + s^2 ) * s' = g*u'/u^3 Integrate both sides to get the answer: g/u^2 + a*(v * sqrt( u^2 + v^2 )/u^2 + arcsinh|v/u|) = const
The kid who solved the projectile problem was Shouryya Ray, born India now living Germany. An abstract of his competition entry (he only won 2nd prize!) was http://jugend-forscht-sachsen.de/2012/teilnehmer/fachgebiet/id/5 [apparently his full paper is not available?] Mitglieder: Shouryya Ray Schule: Martin-Andersen-Nexo-Gymnasium Dresden Analytische Lösung von zwei ungelösten fundamentalen Partikeldynamikproblemen Zwei Probleme aus der klassischen Mechanik haben mehrere Jahrhunderte mathematischer Bemühung getrotzt. Im ersten Problem handelt es sich darum, die Trajektorie eines schräg geworfenen Körpers im erdnahen Schwerefeld und Newton'schen Strömungswiderstand zu berechnen. Das zugrundeliegende Kraftgesetz wurde bereits von Newton (17. Jhd.) entdeckt. Beim zweiten Problem ist das Ziel die Beschreibung einer Partikel-Wand-Kollision unter Hertz’scher Kollisionskraft und linearer Dämpfung. Die Kollisionskraft wurde bereits 1858 von Hertz hergeleitet, eine lineare Dämpfungskraft ist seit Stokes (1850) bekannt. Diese Arbeit setzt sich also die analytische Lösung dieser bisher nur näherungsweise oder numerisch gelösten Probleme zum Ziele. Zunächst werden die beiden Probleme im vollanalytisch gelöst. Für das erste Problem wird zusätzlich anhand der analytischen Lösung das physikalische Verhalten des Systems untersucht und Lösungsskizzen für verallgemeinerte Modelle aufgestellt. Für das zweite Problem wird zwecks Konvergenzsteuerung und Effizienzsteigerung eine semianalytische Optimierung durchgeführt. Schließlich werden die analytischen Ergebnisse mit numerischen Lösungen verglichen, um so Richtigkeit und Konvergenz zu numerisch zu validieren. Entrant: Shouryya Ray School: Martin Andersen Nexo-Gymnasium Dresden Analytical solution of the two unresolved fundamental Particle dynamics problems. Two problems from classical mechanics have defied solution for several centuries. Problem 1: the trajectory of a body thrown at an angle in the near-earth Newton gravitational field and with air drag. The power law was found by Newton (17th century). Problem 2: Describe particle-wall collision under Hertzian collision force and linear damping. The collision energy [as a function of distance of ball-center from wall] was derived in 1858 by Hertz; a linear damping force has been known since Stokes (1850). [If you turn off the damping I guess this is a certain non-linear oscillator.] This paper has the only analytical solution so far, not merely approximate or numerical ones, for the problems solved. The two problems are solved fully analytically. For the first problem, we in addition examine the physical behavior of the system on the basis of the analytical solution and solution sketches drawn up for generalized models. For the second problem a semi-analytical optimization is performed for the purpose of convergence control and efficiency. Finally, the analytical results are compared with the numerical solutions to validate accuracy and numerical convergence. ========================= "Gressen" points out that s = v/u = tan(angle to horizontal of motion) which explains why it is a natural quantity & how Ray thought of using it. "Pelli" notes one can find t as a function of s by integrating dt = -ds / sqrt[C - a*g*(s*sqrt( 1 + s^2 ) + arcsinh|s|)] thus implicitly determining both s' and s as function of t, and then since we know u(s, s') and v(s, s') and hence implicitly u(t) and v(t) we could by a further integration dt deduce projectile position as function of time t. Can any of that be done in closed form? I do not know. But I daresay it could be done out to any number of terms by using infinite series. One obvious move is to regard the air drag coefficient a as small and expand a a perturbative series in a about the known easy no-drag solution. ============= Somebody also pointed out this paper (which I have not seen) K. Yabushita, M. Yamashita, K. Tsuboi: An Analytic Solution of Projectile Motion with the Quadratic Resistance Law Using the Homotopy Analysis Method, Journal of Physics A: Mathematical and Theoretical 40 (2007) 8403-8416.
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Warren Smith