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CBSE Class 11-science Answered

A ship of total mass m is anchored in the middle of a river and water is flowing with a constant velocity v. the horizontal component of the force exerted on the ship by the anchor chain is T. If the anchor chain suddenly breaks, determine the time required for the ship to attain a velocity equal to 0.5v ; assuming that the frictional resistance of the water is proportional to the velocity of the ship relative to water.
Asked by vidyavikram10 | 21 Jul, 2019, 12:57: PM
answered-by-expert Expert Answer
when the chain breaks, force F acting on the ship when it is moving with some instantaneous speed w is given by,
 
F = m (dw/dt)  = T - [ k×(w-v) ]  .....................(1)
 
where k is constant,  v is the speed of flowing water in river and  k(w-v) is the resistive force
due to ship movement in flowing water
 
Hence, we have,  begin mathsize 14px style m space cross times fraction numerator d w over denominator T space plus k v space minus space left parenthesis space k cross times w right parenthesis end fraction space equals space d t end style
begin mathsize 14px style negative m over k space cross times fraction numerator negative k cross times d w over denominator T plus k space v space minus space left parenthesis k cross times w right parenthesis end fraction space equals space d t B y space i n t e g r a t i n g space a b o v e space e q n. comma space space minus integral subscript 0 superscript 0.5 v end superscript m over k space cross times fraction numerator negative k cross times d w over denominator T plus k space v space minus space left parenthesis k cross times w right parenthesis end fraction space space equals space integral subscript 0 superscript t d t minus m over k cross times open square brackets ln space left parenthesis T plus k space v space minus space k cross times w close square brackets subscript 0 superscript 0.5 v end superscript space equals space t m over k cross times ln open parentheses fraction numerator T plus k space v over denominator T space plus space 0.5 space k space v end fraction close parentheses space equals space t end style
Answered by Thiyagarajan K | 21 Jul, 2019, 03:53: PM
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