Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

In Problem 21 it is observed that when the velocity of the sailboat reaches 5 m/sec, the boat begins to rise out of the water and “plane.” When this happens, the proportionality constant for the water resistance drops to b₀ = 60 N-sec/m. Now find the equation of motion of the sailboat. What is the limiting velocity of the sailboat under this wind as it is planning?

The limiting velocity of the sailboat is 10 m/second.
The equation of motion is $x (t) = 10 t + \frac{25}{6} (e^{- 1.2 t} - 1)$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the equation of velocity

There are two forces are acting in the direction of the boat and in the opposite direction of the boat respectively.

$F = 600 N F_{2} = 60 v$

Now

role="math" localid="1664211510034" $m \frac{dv}{dt} = 600 - 60 v m \frac{dv}{dt} = 600 - 60 v \frac{dv}{dt} = 12 - 1.2 v \int \frac{dv}{12 - 1.2 v} = \int dt (Integrating on both sides) - \frac{1}{1.2} \ln |12 - 1.2 v| = t + C \ln |12 - 1.2 v| = - 1.2 t - 1.2 C 12 - 1.2 v = k . e^{- 1.2 t}$

Put v = 5, t = 0 then k = 6

$12 - 1.2 v = 6 . e^{- 1.2 t} v (t) = 10 - 5 . e^{- 1.2 t}$

Step 2: Find the value of equation of motion

$x (t) = \int_{0}^{t} v (s) ds x (t) = \int_{0}^{t} 10 - 5 e^{- 1.2 s} ds = {[10 s + \frac{5 e^{- 1.2 s}}{1.2}]}_{0}^{t} x (t) = 10 t + \frac{25}{6} (e^{- 1.2 t} - 1)$

Hence, The equation of motion is $x (t) = 10 t + \frac{25}{6} (e^{- 1.2 t} - 1)$ .

Step 3: Find the limiting velocity

The limiting velocity of the sailboat is

$\begin{array}{rcl} \lim_{t \to \infty} v (t) & = & \lim_{t \to \infty} (10 - 5^{- 1.2 t}) \\ = & 10 \end{array}$

Hence, the Limiting velocity is 10m/sec.

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Fundamentals Of Differential Equations And Boundary Value Problems

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Step by Step Solution

Step 1: Find the equation of velocity

Step 2: Find the value of equation of motion

Step 3: Find the limiting velocity

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Fundamentals Of Differential Equations And Boundary Value Problems

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