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I'm having difficulty comprehending the response in this solution, regarding the question, 'How large must u be so that the particle will pass through the origin?' Could you please clarify how the solution justifies this aspect?

I'm having difficulty comprehending the response in this solution, regarding the question, 'How large must u be so that the particle will pass through the origin?' Could you please clarify how the solution justifies this aspect?

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter2: Graphical And Tabular Analysis
Section2.1: Tables And Trends
Problem 1TU: If a coffee filter is dropped, its velocity after t seconds is given by v(t)=4(10.0003t) feet per...
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I'm having difficulty comprehending the response in this solution, regarding the question, 'How large must u be so that the particle will pass through the origin?' Could you please clarify how the solution justifies this aspect?

that is, if e6t u - u - -8° Such a value of t will exist if u is such that F(u) > 1, where F FIGURE 5.1 The function F(u). и F = - 2 u - . -8° น The graph of F is shown in Figure 5.1. The condition F > 1 is satisfied if u > 8, but not otherwise. Hence the particle will pass through the origin if u > 8.
Transcribed Image Text:that is, if e6t u - u - -8° Such a value of t will exist if u is such that F(u) > 1, where F FIGURE 5.1 The function F(u). и F = - 2 u - . -8° น The graph of F is shown in Figure 5.1. The condition F > 1 is satisfied if u > 8, but not otherwise. Hence the particle will pass through the origin if u > 8.
Problem 5.4 An overdamped harmonic oscillator satisfies the equation x+10x + 16x = : 0. At time t = O the particle is projected from the point x = 1 towards the origin with speed u. Find x in the subsequent motion. Show that the particle will reach the origin at some later time t if и 2 = e e6t - и 8 How large must u be so that the particle will pass through the origin? Solution The equation of motion is solved in the standard manner by seeking solutions of the form x = eat. Then λ must satisfy the equation the roots of which are λ == 22+102+16 = 0, -2, -8. We have thus found the pair of solutions x = -2t e-8t The general solution of the equation of motion is therefore x = Ae -2t + Be-8t ' where A and B are arbitrary constants. The initial conditions x = 1 and x = -u when t = 0 give the equations A + B = 1, 2A + 8B = u, from which it follows that A = particle is therefore given by - — (u – 8), B = (u 2). The motion of the x = ½ (u − 2)e¯8 – ½ (u — 8)e¯² ¸ The particle is at the origin at time t if - (u− 2)e8t — (u - 8)e¯²t = 0,
Transcribed Image Text:Problem 5.4 An overdamped harmonic oscillator satisfies the equation x+10x + 16x = : 0. At time t = O the particle is projected from the point x = 1 towards the origin with speed u. Find x in the subsequent motion. Show that the particle will reach the origin at some later time t if и 2 = e e6t - и 8 How large must u be so that the particle will pass through the origin? Solution The equation of motion is solved in the standard manner by seeking solutions of the form x = eat. Then λ must satisfy the equation the roots of which are λ == 22+102+16 = 0, -2, -8. We have thus found the pair of solutions x = -2t e-8t The general solution of the equation of motion is therefore x = Ae -2t + Be-8t ' where A and B are arbitrary constants. The initial conditions x = 1 and x = -u when t = 0 give the equations A + B = 1, 2A + 8B = u, from which it follows that A = particle is therefore given by - — (u – 8), B = (u 2). The motion of the x = ½ (u − 2)e¯8 – ½ (u — 8)e¯² ¸ The particle is at the origin at time t if - (u− 2)e8t — (u - 8)e¯²t = 0,
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