Explanation Given: Wind speed is 1 m/sec Constant force on the sailboat due to wind is 600 N Force due to water resistance is proportional to velocity with proportionality constant, b = 100 N-sec/m Mass of sailboat is 50 kg Approach: Force due to wind acting in direction of the velocity of the sailboat and force due to water resistance acting in direction opposite to velocity of the sailboat. Use the 2 nd law of motion to find the differential equation for the motion of the sailboat. As the speed increases the force due to water resistance increases and limiting velocity is attained when force due to wind equals force due to water resistance. Calculation: Let v denote the velocity of the sailboat. Applying 2 nd law of motion we get: m d v d t = 600 − 100 v 50 d v d t = 600 − 100 v d v d t = 12 − 2 v d v d t + 2 v = 12 For this first order linear differential equation the integrating factor is: exp ( ∫ 2 d t ) = exp ( 2 t ) Multiply by the integrating factor to get: d d t ( v exp ( 2 t ) ) = 12 exp ( 2 t ) Integrate with respect to t to get:

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 3.4, Problem 21E

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Limiting velocity of the sailboat.

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Chapter 3.4, Problem 20E

Chapter 3.4, Problem 22E

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Fundamentals of Differential Equations and Boundary Value Problems

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Limiting velocity of the sailboat.

Solution Summary: The author explains that the equation of motion is x(t)=6t+52e-2t

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Limiting velocity of the sailboat.