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Projectile Motion with Resistance and the Lambert W Function
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| The College Mathematics Journal |
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Mathematical investigations of projectile motion have a rich and vital history, going back almost 500 years. Besides the obvious application to ballistics, there is a much more noteworthy connection (for our more pacific purposes) to developments and colorful personalities in mathematics and physics. Many of these ideas are presented in a compelling paper by Groetsch [4], who traces a rich history from Tartaglia to Galileo and then employs a tour de force of undergraduate analysis to answer and expand some classical questions to the case of projectile motion in a resistive medium. The purpose of this paper is to indicate how some of Groetsch's ingenious analysis can be obviated with the help of computer algebra and a recently revived symbolic object, the Lambert W function, that increasingly seems destined for fame and immortality (see FOCUS [1]; Corless et al. [3]). We will use these modern tools to simplify the derivation of some of Groetsch's results while extending them. In particular, we find a symbolic solution for the range as a function of the elevation angle in the presence of a linear resistance. We also give a partial solution to the inverse problem of finding the elevation angles that give rise to given range values by obtaining a closed form for the angle generating the maximum range in terms of the initial velocity and the resistance constant. Two interesting side issues arise in the process of our development. One of them evolves from the need to investigate certain limits involving the Lambert W function. To do this we develop a general theorem, which may be of interest in its own right, about inverse functions arising from real-analytic functions. A second issue relates to the newly emerging area of Experimental Mathematics (Borwein and Corless [2]) and raises practical and philosophical questions about the use of symbolic computation to "discover" new results and the extent to which such computation can be viewed as an accepted form of proof.
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projectile motion, Lambert W, range, resistance, symbolic solution, limit theorem, inverse functions, inverse range, maximum range angle
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| ProjectileMotionwithResistanceJan2004.nb (369.9 KB) - Mathematica Notebook [for Mathematica 5.2] |
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