Interplay between mathematics and physics in science education: didactic approaches to the historical double-cone experiment

The double-cone paradox is a classical experiment widely used in physics education to illustrate the role of the center of mass in apparently counterintuitive motion. Despite its conceptual simplicity, providing a clear and quantitatively accurate explanation of the phenomenon can be challenging, and existing models often rely on advanced mathematical tools or geometrical constructions that are not immediately transparent to students. In this work, we present two new complementary geometrical formulations of the double-cone paradox that do not require calculus and are accessible at the upper high-school level. The first approach is based on an explicit analysis of the motion of the center of mass in the vertical plane containing the bisector of the rails, while the second describes the same phenomenon by directly considering the motion of the contact points along the rails. Both approaches lead to a condition for the occurrence of the paradox that is formally equivalent to those obtained in previous studies. However, the present formulation makes explicit the relation between the actual inclination of the rails and its projection onto the plane of motion, providing a more transparent geometrical interpretation of the phenomenon. This perspective allows the structure of the parameter space to be visualized directly and reveals the existence of a critical condition separating paradoxical and non-paradoxical regimes. The proposed approach provides a simple and effective framework for understanding the double-cone paradox and highlights the role of geometry as a bridge between mathematical formalism and physical intuition.