Module 12  Particle Motion and Parametric Models 
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest 
Lesson 12.2: Harmonic Motion 
Motion along a straight line was illustrated in Lesson 12.1. This lesson will explore harmonic motion, that is, motion that oscillates. Simple harmonic motion is motion that can be modeled by either of the following functions, where t is time: f(t) = a sin (bt + c) + d or g(t) = a cos(bt + c) + d Stating the Problem A particle moves along the vertical line x = 1 according to the equation . Model the motion of the particle from t = 0 to t = 5 seconds using parametric equations. Modeling the Particle's Motion The graph of the following parametric equations illustrates when the particle is moving up, when it is moving down, and when it is at rest.
You should see the ball rise and fall along the vertical line x = 1.
12.2.1 What is the effect of this change on the graph? Another Model Add the following pair of equations to the Y= Editor to produce another model for the particle motion.
12.2.2 Use the Trace feature on the second graph to determine when the particle is moving upward.
12.2.3 When is the particle moving downward?
12.2.4 When is the particle at rest? Graphing the Derivative Add the following equations to the Y= Editor to see the derivative (velocity) of the particle.
12.2.5 Use Trace on the velocity curve to find the first time when the particle is at rest.
12.2.6 What feature of the derivative tells you when the particle is moving up or down? 
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