Question

The area of two holes $A$ and $B$ are $2a$ and $a$ respectively. The holes are at depths $\frac{H}{3}$ and $\frac{2H}{3}$ from the surface of water. Find the correct option(s) -

• A. The velocity of efflux at hole $B$ is $2$ times the velocity of efflux at hole $A$
• B. The velocity of efflux at hole $B$ is $\sqrt{2}$ times the velocity of efflux at hole $A$
• C. The discharge is same through both the holes
• D. The discharge through hole $A$ is $\sqrt{2}$ times the discharge through hole $B$

Answer 1

Answer: (B), (D)

The discharge through hole $A$ is $\sqrt{2}$ times the discharge through hole $B$

Given,
Area of hole $A,A_A=2a$
Area of hole $B,A_B=a$


Velocity of efflux -
For hole $A$ :
$v_A=\sqrt{2g\times \frac{H}{3}}$ ......(1)
$[v=\sqrt{2gh}]$
For hole $B$ :
$v_B=\sqrt{2g\times \frac{2H}{3}}$ ......(2)
From (1) and (2),
$v_B=\sqrt{2}{v}_A$ .......(3)

Discharge of efflux -
For hole $A$ :
$Q_A=2a\times v_A$ ......(4)
$[Q=Av]$
For hole $B$:
$Q_B=a\times v_B=a\times \sqrt{2}{v}_A$ ......(5)
$[$ from (3) $]$
From (4) and (5),
$Q_A=\sqrt{2}{Q}_B$
Hence, option B and D are correct.