Kinematic equations from Taylor series and derivatives beyond acceleration

Abstract

This article proposes a pedagogical approach to teaching kinematics grounded in the Taylor series expansion. We demonstrate how the well-established equations for uniform and uniformly accelerated motion emerge naturally as first and second-order approximations of the position vector's series expansion.
This provides a unified and fundamental origin for these equations that avoids rote memorization, while also offering a practical, in-class application of the Taylor series.
This approach opens the door to exploring the physical meaning of higher-order derivatives of position, such as jerk, snap, crackle, and pop. We trace the history of their nomenclature, propose a standardized terminology for the Portuguese language, and present examples of their applications in engineering and physics. The objective is to offer a perspective that enriches the teaching of kinematics, reinforcing the power of differential calculus as a descriptive and predictive tool.