Modeling Workshop - Explanation of Linearization

These pages were developed to explain some of the problems in linearizing the data gathered from a distance versus time^2 graph. While teaching a block physics/precalculus class, Jim Rynearson and myself have encountered various problems with the linearization method as has been discussed in the listserv. These pages are an attempt to explain from a mathematical perspective the problems with the linearization method. My experience with the method of linearization and physics is very limited, so please feel free to send comments, corrections, and discussions as these pages are my attempt at contributing to the discussion. Send your comments to jwelker@lps.org.

Introduction

The general form of the equation for an object projected vertically upward is d = (-1/2)*a*t^2 + v_i*t + d_i, where "a" is the acceleration due to gravity, "vo" is the initial velocity, and "do" is the initial height. Many have reported errors when these graphs are linearized. The process of linearization replaces the time portion of the graph x-axis with time^2. When the linear regression equation is determined using the distance vs. time^2 data, the graph is in the form of y = m*x + b. However, "x" is now t^2 and y is the distance "d". The resulting equation is d = m*t^2 + b where m is now 1/2 of the accleration due to gravity and b is the initial height. The problem with the linearization method of distance vs time^2 graphs appears to be that it does not factor the initial velocity into the equation. A problem also exists if the intial velocity is zero but the time when the motion begins is not zero. The following pages are write-ups from labs which result in very high error rates with the linearization method. All labs have data recorded using a TI-82 or TI-83 calculator and CBL.

An explanation of this concept follows in the following two labs.

Balldrop Lab - Demonstrates the results of a lab where the time does not start at zero.

Picket Fence Lab - Demonstrates the results of a lab where the initial velocity is not zero.