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Chapter 3: Q3.4-21E (page 117)

A sailboat has been running (on a straight course) under a light wind at 1 m/sec. Suddenly the wind picks up, blowing hard enough to apply a constant force of 600 N to the sailboat. The only other force acting on the boat is water resistance that is proportional to the velocity of the boat. If the proportionality constant for water resistance is b= 100 N-sec/m and the mass of the sailboat is 50 kg, find the equation of motion of the sailboat. What is the limiting velocity of the sailboat under this wind?

Short Answer

Expert verified
  • The equation of motion of sail boat isx(t)=6t+52e-2t-52 .
  • The limiting velocity is 6 m/sec.

Step by step solution

01

Find the equation of velocity 

There are two forces are

F=600F2=-100v

Now

mdvdt=600-100v50dvdt=600-100vdvdt=12-2v∫dv6-v=∫2dt-6ln6-v=t+Cv(t)=6-Ce-2t

Put v(0) = 1 then C = 5

v(t)=6-5e-2t

02

Find the value of equation of motion

x(t)=∫6-5e-2tdtx(t)=6t+5e-2t2+A

When t = 0 then

x(t)=6t+5e-2t2+52

Hence, the equation of motion of sail boat is role="math" localid="1664210783875" x(t)=6t+5e-2t2+52.

The limiting velocity of the sailboat is

limv(t)t→∞=limt→∞(6-5e-2t)=6

Hence, the Limiting velocity is 6 m/sec.

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