Question

Equal volumes of two immiscible liquids of densities $\rho$ and $2\rho$ are filled in a vessel as shown in figure. Two small holes are punched at depth h/2 and 3h/2 from the surface of lighter liquid. If $V_1$ and $V_2$ are the velocities of a flux 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 punches at depth $\frac{h}{2}$ and $\frac{3h}{2}$ 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 from the first hole is(i.e. from surface of remaining liquid)
$h_2=\frac{3h}{2}-\frac{h}{2}=h$
Now, we have the velocity of flux from first hole is
$V_1=\sqrt{2}{h}_{1}g$
$V_1=\sqrt{2}\left(\frac{h}{2}\right)g$
$V_1=\sqrt{h}g$
Also, the velocity of flux from second hole is
$V_2=\sqrt{2}{h}_{2}g$
$V_2=\sqrt{2}hg$
Therefore,
$\frac{V_1}{V_2}=\frac{\sqrt{h}g}{\sqrt{2}hg}$
$\frac{V_1}{V_2}=\frac{1}{\sqrt{2}}$