There are two angles of projection for which the horizontal range is the same. Show that the sum of the maximum heights for these two angles is independent of the angle of projection.

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)`