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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

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

............... (1)

During this time t , boat drifts a distance x due to river flow

..................... (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.

If we equate ( dx/dθ ) = 0 , then we get

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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