Module 15  Particle Motion and Parametric Models  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 15.3: Motion in a Plane  
Motion along a line was modeled in Lesson 15.1 with a polynomial function and Lesson 15.2 modeled harmonic motion along a line with a trigonometric function. In this lesson motion in a plane will be investigated. The Problem A baseball is hit at an angle of 20° above the horizon with an initial velocity of 152 ft/sec and from an initial height of 3 feet off the ground. Will the ball clear a 10foot wall at a distance of 415 feet away from home plate? Modeling the Motion Because the ball's position is restricted to a plane that is vertical with respect to the ground, its motion can be modeled in two dimensions, where x represents the horizontal distance of the ball from home plate and y represents the height of the ball above the ground. The horizontal and vertical positions can each be written as a function of time. The movement of the baseball in the plane can be modeled using the parametric equations below, where
In the given problem v = 152 ft/sec, h = 3 ft, and = 20°. Define the parametric equations so that the graph of the ball's path may be drawn.
15.3.1 Graph the equations and use the Trace feature to decide if the ball clears the 10foot fence that is 415 feet from where it was hit. Click here for the answer. Finding Maximum Height The path of the ball is an inverted parabola, so the maximum height occurs at the vertex, or local maximum. To determine the maximum height of the ball:
The ball reaches its maximum height at approximately t = 1.62 seconds.
15.3.2 Find the maximum height of the ball by evaluating Y_{1T} at the value of t = 1.62. Click here for the answer. 15.3.3 Use the second derivative test to justify that the value found in Question 15.3.2 corresponds to a local maximum. Click here for the answer. 

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