Question

A container of large uniform cross sectional area A resting on a horizontal surface holds two immiscible non-viscous and incompressible liquids of density d and 2d each of height H/2. The lower density liquid is open to the atmosphere having pressure $P_0$. A tiny hole of area a (a<<A) is punched on the vertical side of the lower container at a height $h(0<h<H/2)$ for which range is maximum.

• A. $h=\frac{3H}{8}$
• B. Range $R=\frac{3H}{4}$
• C. Range $R=\frac{3H}{2}$
• D. Velocity of efflux $v$ $=\sqrt{\frac{3gH}{4}}m/s$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $h=\frac{3H}{8}$
B Range $R=\frac{3H}{4}$
D Velocity of efflux $v$ $=\sqrt{\frac{3gH}{4}}m/s$

Applying Bernoulli's equation just inside and just outside the hole,$P_0+\left(\frac{H}{2}\right)\left(d\right)g+(\frac{H}{2}-h)\left(2d\right)\left(g\right)=\frac{1}{2}\left(2d\right)v^2+P_0$

$\therefore v=\sqrt{\frac{g}{2}(3H-4h)}$


$t=\sqrt{\frac{2h}{g}}s$

$\therefore$ $R=vt=\sqrt{(3H-4h)h}$

$R$ will be maximum at $h=\frac{3H}{8}$.
$R_{max}=\frac{3H}{4}m$

Velocity of efflux $v$ $=\sqrt{\frac{3gH}{4}}m/s$