Program Explanation Given information: The graph of the velocity function of a particle that travels along the straight line is redrawn as shown in Figure 1. Graph: It is observed from Figure 1 that the particle starts from the origin and in the interval of time [ 0 , 3 ] the velocity function is in linear form. Therefore, the velocity function is t The relation between velocity function and the position function can be expressed as v ( t ) = d x d t Therefore, the position function can be expressed as x ( t ) = ∫ v ( t ) d t = ∫ 5 3 t d t = 5 3 t 2 2 + C = 5 t 2 6 + C Therefore, the position function is x ( t ) = 5 t 2 6 + C . The particle starts at the origin. Therefore, the initial position x ( 0 ) = 0 . Now apply the condition of the initial position x 0 = 0 to obtain the value of C . 0 = 5 ( 0 ) 2 6 + C C = 0 Substitute 0 for C in the equation x ( t ) = 5 t 2 6 + C to obtain the particular solution. x ( t ) = 5 t 2 6 + 0 x ( t ) = 5 t 2 6 Therefore, the position function is x ( t ) = 5 t 2 6 in the interval of time [ 0 , 3 ] . The velocity function is v ( t ) = 5 in the interval of time [ 3 , 7 ] . Therefore, the position function can be expressed as x ( t ) = ∫ v ( t ) d t = ∫ 5 d t = 5 t + C Since the position function is x ( t ) = 5 t 2 6 in the interval of time [ 0 , 3 ] . Substitute 3 for t in equation x ( t ) = 5 t 2 6 to obtain the corresponding value of x ( 3 ) . x ( 3 ) = 5 ( 3 ) 2 6 x ( 3 ) = 15 2 Therefore, the value of x ( 3 ) is 7.5 . Now apply the condition ofthe initial position x ( 3 ) = 7.5 to obtain the value of C . 7.5 = 5 ( 3 ) + C 7.5 = 15 + C − 7.5 = C Substitute − 7.5 for C in the equation x ( t ) = 5 t + C to obtain the particular solution. x ( t ) = 5 t − 7.5 Therefore, the position function is x ( t ) = 5 t − 7.5 in the interval of time [ 3 , 7 ] . The velocity function is v ( t ) = − 5 t 3 + 50 3 in the interval of time [ 7 , 10 ] . Therefore, the position function can be expressed as x ( t ) = ∫ v ( t ) d t = ∫ ( − 5 t 3 + 50 3 ) d t = − 5 t 2 6 + 50 3 t + C Since the position function is x ( t ) = 5 t − 7

5th Edition

ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

Question

Chapter 1.2, Problem 22P

Program Plan Intro

To sketch: The graph of the position function x(t) for 0≤t≤10 .

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Chapter 1.2, Problem 21P

Chapter 1.2, Problem 23P

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Chapter 1 Solutions

EBK DIFFERENTIAL EQUATIONS

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The graph of the position function x ( t ) for 0 ≤ t ≤ 10 .

Solution Summary: The author illustrates the graph of the velocity function of a particle that travels along the straight line as shown in Figure 1.

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To sketch: The graph of the position function x(t) for 0≤t≤10 .

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Chapter 1 Solutions