Another Broken Symmetry
Author: Chuck W. Groetsch
The College Mathematics Journal, Vol. 36, No. 2 (March, 2005), pp. 109-113
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/30044833
Tag: Higher order linear equations and linear systems
Abstract: This paper acknowledges the beautiful symmetries that nature bears even in the most simplistic phenomena. In non-resistive projectile motion, there are spatial and temporal symmetries that occur. For example, the altitude of the projectile is the same for equidistant horizontal displacements to the left and right of the apex and the time of ascent from the launching position to the apex is the same as the time of descent from the apex to the impact location. However, it is well known that the path of a super-sonic, long-range missile does not have these symmetries. They are broken by the addition of linear air resistance to the Galilean model. The author chooses to prove one of the asymmetries but makes reference to other papers that show how the other symmetries are broken. In this new model, the vertical and horizontal motion of the projectile can be expressed as second order differential equations. Integrating the equations and adding in parameters of the projectile allows the author to plot the trajectory of the projectile. The path of the projectile is not symmetric about the apex. In fact, the altitude of the projectile is greater during the ascent for equidistant horizontal displacements to the left and right of the apex.
Review by Hunter C. Sullivan and Rebecca Woods, Stephen F. Austin State University, May 5, 2015.