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Question

A, B, C, D are four trees, located at the vertices of a square. Wind blows from A to B with uniform speed. The ratio of times of flight of a bird from A to B and from B to A is 'n'. At what angle should the bird fly from the direction of wind flow, in order that it starts from A and reaches C.Let


A

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B

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C

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D

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Solution

The correct option is D


Let velocity of bird w.r.t air is v and velocity of wind is u.

Velocity of bird from A to B |vAB|=v+u

Velocity of bird from B to A |vBA|=vu,tAB=1v+u,tBA=1vu

tABtBA=nn=vuv+u,uv=1n1+n ...................(i)

Wind velocity vw=u^i

Velocity of bird w.r.t. ground vb will be in direction of line joining AC

Vb=vb cos 45 ^i+vb sin 45 ^j

vb=vb2^i+vb2^j

vbw=vbvw

Drawing the vector diagram

vbw=v cosα^i+v sinα^j

vb=vbw+vw

vb2^i+vb2^j=v cosα^i+v sinα^j+v^i

Equation ^i and ^j components

vb2=(v cosα+u) ................(i)

vb2=v sinα ................(ii)

Equation equation (i) and (ii) we get

vcosα+u=v sinα

u(cosαsinα)=u

sinαcosα=uv

sinαcosα=1n1+n

Squaring,

12 sin α cos α=(1n)2(1+n)2

Sin 2α=4n(1+n)2

α=12sin1[4n(1+n)2]


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