Q) Two stones are projected so as to reach the same distance from the point of projection on a horizontal plane. The maximum height reached by one exceeds the other by an amount equal to half the sum of the heights attained by them. The angles of projection for stones are?

A)
Since Range = ucos@ . 2usin@/g = u^2.sin2@/g
therefore for angle of projection @ and 90-@, range is same.
so the angles are X, and 90 - X

Height = u^2.sin^2@ / 2g
H - h = 1/2 * (H +h)
H - h = H/2 +h/2
H/2 = 3 h / 2
H = 3h
H/h = 3 = (u^2.sin^2 X / 2g ) (2g / u^2. sin^2 Y)
3 = sin^2 X/sin^2 Y
3 = sin^2 X / sin^2 (90 - X) = tan^2 X
tan X = root (3)
X = 60 degrees
and Y = 30 degrees

So the angles of projection are 30 and 60 degrees with the horizontal.

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