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
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|=v−u,tAB=1v+u,tBA=1v−u
tABtBA=n⇒n=v−uv+u,uv=1−n1+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=vb√2^i+vb√2^j
→vbw=→vb−→vw
Drawing the vector diagram
⇒→vbw=v cosα^i+v sinα^j
→vb=→vbw+→vw
vb√2^i+vb√2^j=v cosα^i+v sinα^j+v^i
Equation ^i and ^j components
vb√2=(v cosα+u) ................(i)
vb√2=v sinα ................(ii)
Equation equation (i) and (ii) we get
vcosα+u=v sinα
u(cosα−sinα)=−u
sinα−cosα=uv
sinα−cosα=1−n1+n
Squaring,
1−2 sin α cos α=(1−n)2(1+n)2
Sin 2α=4n(1+n)2
α=12sin−1[4n(1+n)2]