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 determine the speed of the liquid coming out of the hole in the vessel and identify which quantity it is independent of.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a liquid of density \( \rho \) filled in a vessel to a height \( H \).
- There is a hole with a cross-sectional area \( A \) at a depth \( h \) below the free surface of the liquid.

2. **Apply Torricelli's Law**:
- According to Torricelli's law, the speed \( v \) of the liquid flowing out of the hole can be derived from the principle of conservation of energy. The speed is given by:
\[ v = \sqrt{2gh} \]
- Here, \( g \) is the acceleration due to gravity, and \( h \) is the vertical distance from the liquid surface to the hole.

3. **Identify the Variables**:
- The speed \( v \) depends on:
- \( g \) (acceleration due to gravity)
- \( h \) (depth of the hole)
- \( A \) (cross-sectional area of the hole)
- \( \rho \) (density of the liquid)

4. **Analyze the Dependence**:
- From the equation \( v = \sqrt{2gh} \), we can see that the speed \( v \) is directly influenced by \( g \) and \( h \).
- The cross-sectional area \( A \) of the hole does not appear in the equation for \( v \) when considering the speed derived from Torricelli's law.

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

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