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Graphical Analysis and Equations of Uniformly Accelerated Motion - A Unified Approach

Introduction

How do we teach physics?

Sometimes looking at the textbooks we use can be revealing. While individual authors would undoubtedly protest, there are as many common features in textbooks as there are unique ones. This is especially true concerning the teaching and study of kinematics. To simplify the discussion, it is possible to break textbooks into three general categories: calculus-based, algebra-based and conceptual.

Calculus-based textbooks, often given titles similar to “University Physics” or “Physics for Scientists and Engineers”, typically approach a description of motion using differentiation and assume that readers already have some familiarity with calculus. While this is a powerful approach that is broadly applicable for studying a wide range of motion, the ultimate result is most frequently the study of uniformly accelerated linear motion. Not that this is bad—many interesting situations can be successfully modeled by this approximation and the required manipulations are readily accessible to beginning students of calculus. Interestingly, algebra- based textbooks, given titles such as “College Physics” or just ”Physics”, while necessarily forgoing a description of motion involving calculus, typically arrive at the same study of uniformly accelerated linear motion. In these algebra-based texts the development of the defining motion relationships often evolves using seemingly ad hoc, logical justification. For example, the idea that the distance traveled is equal to the average speed multiplied by the time of travel is combined with the statement that for uniform acceleration the average speed is just half the sum of the beginning and ending speeds to arrive at one of the underlying equations describing uniformly accelerated motion. Conceptual textbooks, by their very nature, do not necessarily provide a comprehensive, equation-based description even of uniformly accelerated motion.

An important pedagogical advance in instruction of motion is the use of motion detectors in calculator or computer-based explorations1,2,3,4. Such an approach allows even students with no calculus background to explore the relationships among position, velocity and acceleration versus time graphs because the calculator or computer software automatically generates the correct, calculus-based relationships. While it is possible for a computer to manipulate seemingly complex graphs with apparent ease, when it is time for students to mimic those manipulations themselves they will typically be reduced to dealing with situations where the resulting velocity and acceleration vs. time graphs are piecewise linear and the regions between the graphs and the time axis are rectangular, triangular, or trapezoidal. Almost by default, we are brought back to exactly the same position of exploring uniformly accelerated linear motion. The

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Turner, W., & Ellis, G. (2009, June), Graphical Analysis And Equations Of Uniformly Accelerated Motion: A Unified Approach Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. 10.18260/1-2--4947