Question

Equal volume of two immiscible liquids of densities $\rho$ and $2\rho$ are filled in a vessel as shown in the figure. Two small holes are punched at depths $h/2$ and $3h/2$ from the surface of the lighter liquid. If $v_1$ and $v_2$ are the velocities of efflux at these two holes, then $v_1/v_2$ is

• A. $\frac{1}{2\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is D $\frac{1}{\sqrt{2}}$

Here, two small holes are punched at depth of $h/2$​ and $3h/2$​ from the surface of lighter liquid.
Hence, the height of first hole from the surface of lighter liquid is
$h_1=\frac{h}{2}$​
and the height of second hole surface of lighter liquid
$h_2=3h/2$
Now, we have the velocity of flux from first hole is
$v_1=\sqrt{2h_{1}g}$

$v_1=\sqrt{2g\left(\frac{h}{2}\right)}=\sqrt{gh}$ ... (i)

Velocity of efflux at second hole can be found from Bernoulli's theorem,

$\rho gh+2\rho g\left(\frac{h}{2}\right)=\frac{1}{2}\left(2\rho \right)v_2^2$

$\Rightarrow v_2=\sqrt{2gh}$ ... (ii)

$\therefore \frac{v_1}{v_2}=\frac{1}{\sqrt{2}}$