A,B,C, and D are four trees, located at the vertices of a square (Fig. 5.117). 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 (a) reaches C, (b) reaches D ?

A,B,C, and D are four trees, located at the vertices of a square (Fig.

1.	A,B,C, and D are four trees, located at the vertices of a square (Fig. 5.117). 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 (a) reaches C, (b) reaches D ?
Answer» Solution :Let the velocity of bird w.r.t. air is `v` and the velocity of wind is `u`. The velocity of bird from `A` to `B`, `\|vec v_(A B)\| = v + u` Velocity of bird from `B` to `A`, `\|vec v_(B A)\| = v - u` `t_(A B) = (l)/(v + u), t_B A = (l)/(v - u)` `t_(A B)/(t_(B A)) = N rArr n = (v -u)/(v + u)` `(u)/(v) = (1 - n)/(1 + n)` ...(i) (a) MOTION of bird from `A to C` `vec v_("bird") = vec v_("bird, wind") + vec v_("wind") = vec v + vec u` From velocity diagram, we can use sine rule : `(u)/(sin theta) = (v)/(sin 45^@)` `rArr sin theta = (u)/(sqrt(2) v) = (1 - n)/(sqrt(2) (1 + n)` `sin (prop - 45^@) = (1 - n)/(sqrt(2)(1 + n)` `sin prop COS 45^@ - cos prop sin 45^@ = (1 - n)/(sqrt(2)(1 + n)` `sin prop - cos prop = (1 - n)/(1+ n)` SQUARING both sides, `1 - 2 sin prop cos prop = ((1 - n)/(1 + n))^2` `sin 2 prop = (4 n)/((1 + n)^2)` `rArr prop = (1)/(2) sin^-1 [(4 n)/((1 + n)^2)]` (b) For motion of bird from `A` to `D` : From vector diagram, it is CLEAR `sin theta' = (u)/(v) = (1 - n)/(1 + n)` `sin( prop - 90^@) = (1 - n)/(1 + n)` `cos prop = - ((1 - n))/(1 + n)` `prop = cos^-1 [(-(1 - n))/((1 + n))]`. . (a) . (b) .