A liquid of density `rho` is filled in a vessel up to height H and a hole of cross section area A is made at a depth h below the free surface of liquid. The speed of liquid coming out of the hole is independent of

Options

A. h
B. a
C. `rho`
D. g

Answer

To solve the problem, we need to analyze the situation described in the question. We have a liquid of density \( \rho \) in a vessel, and a hole is made at a certain depth below the surface. We want to find out which quantity the speed of the liquid coming out of the hole is independent of.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- We have a liquid with density \( \rho \) filled to a height \( H \) in a container.
- A hole with cross-sectional area \( A \) is located at a depth \( h \) below the liquid's surface.

2. **Applying Torricelli's Law**:
- According to Torricelli's law, the speed of efflux \( v \) of a liquid through a hole under the influence of gravity can be expressed as:
\[ v = \sqrt{2gh} \]
- Here, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column above the hole.

3. **Identifying the Variables**:
- The speed \( v \) depends on:
- \( g \) (acceleration due to gravity)
- \( h \) (height of the liquid column)
- The cross-sectional area of the hole \( A \) and the cross-sectional area of the container \( A_c \) can also influence the flow rate but do not directly affect the speed of efflux.

4. **Analyzing the Independence**:
- Now, we need to identify which quantity the speed \( v \) is independent of.
- The density \( \rho \) of the liquid does not appear in the formula for \( v \). This means that the speed of the liquid coming out of the hole does not depend on the density of the liquid.

5. **Conclusion**:
- Therefore, the speed of the liquid coming out of the hole is independent of the density \( \rho \).

### Final Answer:
The speed of the liquid coming out of the hole is independent of the density \( \rho \).