Taylor Series Kinematics

TL;DR

This paper reveals that basic kinematic equations are Taylor series expansions, offers derivations including higher derivatives like jerk, and suggests integrating this perspective into physics education to enhance understanding.

Contribution

It introduces a Taylor series-based derivation of kinematic equations, including higher derivatives, and provides teaching strategies and resources for physics instruction.

Findings

Kinematic equations are Taylor series expansions.

Derived generalized x(t) including jerk and higher derivatives.

Proposed integrating Taylor series kinematics into introductory physics courses.

Abstract

Has it ever occurred to you that the kinematic equations for uniformly accelerated one-dimensional motion are Taylor series expansions? If not, you are in good company. I didn't know this myself until a colleague pointed it out to me many years ago, and I was stunned to learn something new and wonderful about something so familiar. Accordingly, my first objective in this paper is to clearly present the not-widely-known Taylor series derivations of these basic equations to a population primed to deeply appreciate them: people, like me, who teach introductory physics. Following this, I use the Taylor series approach to derive a generalized one-dimensional expression for x(t) that includes the jerk and further kinematic time derivatives, which have importance in many real-world applications and in which there has been renewed pedagogical interest. I also outline teaching suggestions and provide student-accessible video derivations to support instructors who would like to incorporate Taylor series kinematics into their teaching, while identifying sequencing-related challenges. I close with the observation that the traditional second calculus course, which is largely free of sequencing issues, could be a great place to incorporate and leverage Taylor series kinematics, and I briefly outline an early-stage pilot collaboration to explore this possibility.