A particle start from rest, with uniform acceleration at time `t`, choose the correct option

Answer

`v_(0)=sqrt(gH(1+l^(1)/H^(2))), tan alpha=H/l`

Explanation

`H/2` distance covered by free falling body
`H/2=1/2 g t^(2), t=sqrt(H/g)`

In same time, projectile also travel vertical distance `H/2`, then `H/2=v sin alpha sqrt(H/2)-1/2gH/g`
`v sin alpha=sqrt(gH)` …(i)
also `l=v cos alpha sqrt(H/g), v cos alpha=lsqrt(g/H)` ...(ii)
From equation (i) and (ii)
`tan alpha=H/l v^(2) sin^(2) alpha+v^(2) cos^(2) alpha=gH+l^(2) g/H`
`v=sqrt(gH(1+l^(2)/H^(2))t)`