two particles are projected upwards with the same initial velocity `v_(0)` in two different angles of projection such that their horizontal ranges are the same. The ratio of the heights of their horizontal ranges are the same. The ratio of the heights of their highest point will be

Options

A. `tan^(2) theta_(1)`
B. `v_(0)^(2) sin^(2) theta_(1)`
C. `v_(0) sin theta_(1)`
D. `v_(0)//cos theta_(1)`

Answer

A

Explanation

As the horizontal ranges are the same.
`:. (v_(0)^(2)sin 2 theta)/(g) = (v_(0)^(2)sin 2 theta_(2))/(g)`
So, `sin 2theta_(1) = sin 2theta_(2)`
or `2 theta_(1) = pi - theta_(2)`
`rArr theta_(1) +theta_(2) = pi//2`
`:. (h_(1))_(max) = (v_(0)^(2)sin^(2)theta_(1))/(2g)`
and `(h_(2))_(max) = (v_(0)^(2)sin^(2)theta_(2))/(2g)`
`:. ((h_(1))_(max))/((h_(2))_(max)) =(sin^(2)theta_(1))/(sin^(2)theta_(1)) =(sin^(2)theta_(1))/(cos^(2)theta_(1)) = tan^(2)theta_(1)`