Explanation Given: Constant force on the sailboat due to wind is 600 N Force due to water resistance is proportional to velocity with proportionality constant, b = 60 N-sec/m Mass of sailboat is 50 kg Boat starts to plane when it reaches 5 m/sec 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. Use the instant when boat begins to “plane” as the initial condition. Calculation: Let v denote the velocity of the sailboat. Applying 2 nd law of motion we get: m d v d t = 600 − 60 v 50 d v d t = 600 − 60 v d v d t = 12 − 6 5 v d v d t + 1.2 v = 12 For this first order linear differential equation the integrating factor is: exp ( ∫ 1.2 d t ) = exp ( 1.2 t ) Multiply by the integrating factor to get: d d t ( v exp ( 1.2 t ) ) = 12 exp ( 1.2 t ) Integrate with respect to t to get: v exp ( 1.2 t ) = 10 exp ( 1

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 22E

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Limiting velocity of the sailboat.

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Chapter 3.4, Problem 21E

Chapter 3.4, Problem 23E

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(b) Let I[y] be a functional of y(x) defined by [[y] = √(x²y' + 2xyy' + 2xy + y²) dr, subject to boundary conditions y(0) = 0, y(1) = 1. State the Euler-Lagrange equation for finding extreme values of I [y] for this prob- lem. Explain why the function y(x) = x is an extremal, and for this function, show that I = 2. Without doing further calculations, give the values of I for the functions y(x) = x² and y(x) = x³.

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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 how the equation of motion is x=10t+256left, and the limiting velocity 10m/sec.

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Limiting velocity of the sailboat.