Another Broken Symmetry

Abstract

Galileo's flat earth model for the motion of an unresisted projectile is commonly considered to be the genesis of mathematical physics. The simplicity and elegance of the model has great appeal and its analysis reveals a triad of beautiful symmetries. First there is the symmetry of the trajectory itself: the graceful arc of the trajectory is a parabola whose axis is the perpendicular bisector of the horizontal range of the projectile. There is also a temporal symmetry: the time of ascent from the launch point to the apex is the same as the time of descent from the apex to the impact point. Finally, there is a symmetry of inclination: two shots with launch angles symmetrically placed with respect to the 450 launch angle have equal horizontal ranges. Resistance is the enemy of symmetry. Various asymmetries associated with air resistance have been recognized since the beginnings of the scientific study of ballistics in the early sixteenth century. For example, da Vinci's conjectured trajectories (see, for example, [7]) and Tartaglia's flawed pseudo-analytic trajectories [2] display marked asymmetries. In the first half of the seventeenth century, Marin Mersenne and Rene Descartes performed experiments that showed an asymmetry of the ascent and descent times for a projectile in air (see [4] and the references cited therein). A rather cryptic remark by Edmund Halley [6] in the latter part of the seventeenth century indicated that he observed a systematic asymmetry in the horizontal range when the firing angles are symmetric with 450. The author has found that the study of the simplest model for resistance, that in which the resistive force is proportional to the velocity vector, is an excellent vehicle to illustrate a number of topics in elementary and intermediate mathematical analysis. Several notions from first-year calculus, fixed point theorems, implicit function theorems, and various aspects of modeling and computation based on the linear resistance model, have been illustrated by the author in classes in honors calculus, intermediate analysis, numerical analysis and mathematical modeling (see [3, Ch. 4, Sect. 2] for some examples). In [4] the nature of the asymmetry in the time of ascent-descent for the linear resistance model is explored, and the type of asymmetry of the range function with respect to the firing angle suggested by Halley's observation is rigorously established in [5]. In this note we relate how suggestions made by members of a class in mathematical modeling for in-service high school teachers led to a rigorous proof of a specific asymmetry in the trajectory in the linear resistance model. We first review the well known symmetries in the classical model.