Dimensional analysis is often a subject reserved for students of fluid mechanics. However, the principles of scaling and dimensional analysis are applicable to various physical problems, many of which can be introduced early on in a university physics curriculum. Here, we revisit one of the best-known examples from a first course in classic mechanics, namely the falling ball problem: a ball is thrown with an initial velocity and height while experiencing gravity and viscous drag. We treat two representative cases of drag forces, one linear and one quadratic in velocity. We demonstrate that the ball's motion is governed by two dimensionless parameters: (i) a Froude number ("Fr") comparing the ball's initial kinetic to potential energy and (ii) a drag coefficient (C[subscript D]) comparing the initial drag force to the ball's own weight. By investigating extreme, yet simple hypothetical cases for "Fr" and C[subscript D], we demonstrate how students can grasp the role of the parameters relating the ball's initial conditions in governing several physical behaviors displayed by the system. Advocating early on exposure to dimensional analysis is undoubtedly beneficial in building physical intuition, but also it illustrates how physical systems characterized by many variables may be assimilated by reducing their inherent complexity.