Two particles A and B are projected from the same point with the same velocity of projection but at different angles `alpha `and `beta` of projection, such that the maximum height of A is two-third of the horizontal range of B. then which of the following relations are true?

Options

A. range of `A`= maximum height of `B`
B. `3(1-cos 2 alpha)=8 sin 2 beta`
C. maximum value of `beta` is `sin^(-1)(3//4)`
D. maximum horizontal range of `A` =`u^(2)//g` and this occurs when `beta=1/2sin^(-1)(3/8)`

Answer

B, D

Explanation

`(v^(2)sin^(2)alpha)/(2g)=2/3U^(2)(sin 2beta)/gsin^(2)alpha=4/3sin2beta`
`0le sin^(2) alpha le 1 , 0 le sin2beta le 1 , beta le sin^(-1)(3/4)`
`R_(A)=(U^(2) sin 2alpha)/g`
`3sin^(2)alpha=4sin2beta , R_(max) "at" alpha=T/4`
`3(1-cos2alpha)=8sin2beta , beta=1/2sin^(-1)(3/8)`
`R_(A) ne H_(max).B`