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a boat moves with velocity which is n times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting.    I din understand how dx/d theta=ud/v sec theta tan theta-d sec^2 theta like whr did the addition sec theta come from??? so pls explain tht
Asked by joanmaria916 | 17 Jul, 2023, 11:24: AM
answered-by-expert Expert Answer
Let us consider that boat moves with velocity v in a direction that makes angle θ with a line along the breadth of river as shown in figure. 
 
Velocity v of boat can be resolved as (i) ( v cosθ ) along the line parallel to breadth of river and ( v sinθ ) parallel to direction of river flow velocity u. 
 
Time taken t to cross the river is
 
 begin mathsize 14px style t space equals space fraction numerator d i s tan c e over denominator v e l o c i t y end fraction space equals space fraction numerator l over denominator v space cos theta end fraction end style ............... (1) 
 
During this time t , boat drifts a distance x due to river flow 
 
begin mathsize 14px style x space equals space v e l o c i t y space cross times space t i m e space equals space left parenthesis space u space minus space v space sin theta space right parenthesis space cross times space fraction numerator l over denominator v cos theta end fraction end style ..................... (2)
It can be seen from eqn.(2) , drift is a function of angle θ . 
 
Minimum drift can be determined by differentiating eqn.(2) with respect to θ and equating the derivative to zero.
 
begin mathsize 14px style fraction numerator d x over denominator d theta end fraction space equals l over v open parentheses fraction numerator u minus v space sin theta over denominator space cos theta end fraction close parentheses space equals space l over v space fraction numerator d over denominator d theta end fraction open parentheses u space s e c theta space minus space v space tan theta close parentheses end style
 
begin mathsize 14px style fraction numerator d x over denominator d theta end fraction space equals space l over v open parentheses u space s e c theta space tan theta space minus space v space s e c squared theta close parentheses end style
If we equate ( dx/dθ ) = 0 , then we get
 
begin mathsize 14px style left parenthesis space u space s e c theta space tan theta space minus space v space s e c squared theta space right parenthesis space equals space 0 end style
 
begin mathsize 14px style left parenthesis space u space cross times fraction numerator 1 over denominator cos theta end fraction space cross times fraction numerator sin theta over denominator cos theta end fraction space minus space v space fraction numerator 1 over denominator cos squared theta end fraction space right parenthesis space equals space 0 end style
 
begin mathsize 14px style open parentheses u space sin theta space minus space v close parentheses space equals space 0 end style
From above expression, we get , sinθ = (v/u) ,
 
Hence to get minimum drift , angle θ should be , θ = sin-1 ( v/ u ) 
 
If boat speed v = u/n , then θ = sin-1 ( 1/n )

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