Question

A wide vessel with a small hole at the bottom is filled with two liquids. The density and height of one liquid are ${\rho }_{1}and{h}_1$ and that of the other are ${\rho }_2$ and $h_2$. (Given ${\rho }_1>{\rho }_2)$. The velocity of liquid coming out of the hole is :

• A. $v=\sqrt{2g(h_1+h_2)}$
• B. $v=\sqrt{2g(h_1{\rho }_1+h_2{\rho }_2)/({\rho }_1+{\rho }_2)}$
• C. $v=\sqrt{2g[h_1+\frac{h_2{\rho }_2}{{\rho }_1}]}$
• D. $v=\sqrt{2g[\frac{h_1{\rho }_1}{{\rho }_2}+h_2]}$

Answer 1

The correct option is C $v=\sqrt{2g[h_1+\frac{h_2{\rho }_2}{{\rho }_1}]}$

Applying Bernoulli's equation to points A and B
$P_0+{\rho }_{1}g{h}_1+{\rho }_{2}g{h}_2+0=P_0+\frac{1}{2}{\rho }_1{v}^2$
where v is the velocity at point B
$\therefore \frac{1}{2}{\rho }_1{v}^2=g({\rho }_1{h}_1+{\rho }_2{h}_2)$

$\therefore v_1^2=2g\frac{({\rho }_1{h}_1+{\rho }_2{h}_2)}{{\rho }_1}=2g(h_1+\frac{{\rho }_2}{{\rho }_1}\times h_2)$

$\therefore v_1=\sqrt{2g(h_1+\frac{{\rho }_2}{{\rho }_1}{h}_2)}$