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Find parametric equations for the path of a particle that moves along the curve y = 5 − 4x² from (1, 1) to (-1, 1), and back to (1, 1) as 0≤t≤1.

Find parametric equations for the path of a particle that moves along the curve y = 5 − 4x² from (1, 1) to (-1, 1), and back to (1, 1) as 0≤t≤1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Find parametric equations for the path of a particle that moves along the curve \( y = 5 - 4x^2 \) from \( (1, 1) \) to \( (-1, 1) \), and back to \( (1, 1) \) as \( 0 \leq t \leq 1 \). **Explanation:** To solve this problem, the goal is to express both \( x \) and \( y \) as functions of a parameter \( t \). - The curve given is \( y = 5 - 4x^2 \). - The particle moves from \( x = 1 \) to \( x = -1 \) and back. **Steps to Find Parametric Equations:** 1. **Parameterize \( x(t) \):** - Since \( x \) moves from 1 to -1 and back, a cosine or sine function can be used. Let’s use the cosine function for simplicity. - Set \( x(t) = \cos(\pi t) \). - At \( t = 0 \), \( x(0) = \cos(0) = 1 \). - At \( t = 0.5 \), \( x(0.5) = \cos(\pi \times 0.5) = -1 \). - At \( t = 1 \), \( x(1) = \cos(\pi) = 1 \). 2. **Determine \( y(t) \) Using the Curve:** - Using the equation \( y = 5 - 4x^2 \), substitute \( x(t) \). - \( y(t) = 5 - 4(\cos(\pi t))^2 \). These parametric equations \( x(t) = \cos(\pi t) \) and \( y(t) = 5 - 4(\cos(\pi t))^2 \) describe the movement of the particle along the given path over the interval \( 0 \leq t \leq 1 \).
Transcribed Image Text:**Problem Statement:** Find parametric equations for the path of a particle that moves along the curve \( y = 5 - 4x^2 \) from \( (1, 1) \) to \( (-1, 1) \), and back to \( (1, 1) \) as \( 0 \leq t \leq 1 \). **Explanation:** To solve this problem, the goal is to express both \( x \) and \( y \) as functions of a parameter \( t \). - The curve given is \( y = 5 - 4x^2 \). - The particle moves from \( x = 1 \) to \( x = -1 \) and back. **Steps to Find Parametric Equations:** 1. **Parameterize \( x(t) \):** - Since \( x \) moves from 1 to -1 and back, a cosine or sine function can be used. Let’s use the cosine function for simplicity. - Set \( x(t) = \cos(\pi t) \). - At \( t = 0 \), \( x(0) = \cos(0) = 1 \). - At \( t = 0.5 \), \( x(0.5) = \cos(\pi \times 0.5) = -1 \). - At \( t = 1 \), \( x(1) = \cos(\pi) = 1 \). 2. **Determine \( y(t) \) Using the Curve:** - Using the equation \( y = 5 - 4x^2 \), substitute \( x(t) \). - \( y(t) = 5 - 4(\cos(\pi t))^2 \). These parametric equations \( x(t) = \cos(\pi t) \) and \( y(t) = 5 - 4(\cos(\pi t))^2 \) describe the movement of the particle along the given path over the interval \( 0 \leq t \leq 1 \).
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