Question

In a cylindrical vessel containing liquid of density $\rho$, there are two holes in the side wall at heights of $h_1$ and $h_2$ respectively such that the range of efflux at the bottom of the vessel is same. The height of a hole, for which the range of efflux would be maximum will be

• A. $h_2-h_1$
• B. $h_2+h_1$
• C. $\frac{h_2-h_1}{2}$
• D. $\frac{h_2+h_1}{2}$

Answer 1

The correct option is D $\frac{h_2+h_1}{2}$

We know that range of efflux for hole at height $h$ is
$R=2\sqrt{(H-h)h}$
Given range at both heights $h_1,h_2$ are same, so
$2\sqrt{(H-h_1)h_1}=2\sqrt{(H-h_2)h_2}$
$(H-h_1)h_1=(H-h_2)h_2$
$H{h}_1-h_1^2=H{h}_2-h_2^2$
$H(h_1-h_2)=h_1^2-h_2^2=(h_1+h_2)(h_1-h_2)$
$\therefore H=(h_1+h_2)$
For maximum range, $\frac{dR}{dh}=0$
i.e $\frac{2}{2\sqrt{(H-h)h}}\times (H-2h)=0$
$\Rightarrow h=\frac{H}{2}$
So, $h=\frac{H}{2}=\frac{(h_1+h_2)}{2}$