a stone is thrown in such a manner that it would just hit a bird at the top of a tree and afterwards reach a maximum height double that of the tree . if at the moment of throwing the stone the bird flies away horizontally with constant velocity and the stone hits the bird after some time . the ratio of horizontal velocity of stone to that of the bird is 1¡n+1¡¡Ìn?
1 answer

Well, to make it easy, let's say the tree is 50 meters high. Then the max stone height will be 100 meters.
That gives us a vertical problem to solve.
What is Vi, initial vertical velocity?
(1/2) m Vi^2 = m g h
so
Vi = sqrt(2 g h = sqrt (200*9.81) = 44.3 m/s
so when will it get to the tree top?
h = Vi t  4.9 t^2
50 = 44.3 t  4.9 t^2
4.9 t^2 44 t + 50 = 0
t = [ 44 +/ sqrt(1936 980) ] /9.81
t = [ 44 +/ 30.9 ] /9.81
t = 1.34 seconds on the way up, WHERE IT WOULD HAVE HIT THE BIRD
t = 7.64 seconds on the way down  WHEN IT HITS THE BIRD!
the bird flew for 7.64 seconds at speed u so went 7.64 u meters from the tree
The stone went at horizontal speed s so went a total of 7.64 s meters
of which 1.34 s was the distance from the thrower to the tree. so (7.641.34) s = 7.64 u
I guess you can find s/u now :)