A standing tuning fork of frequency `f` is used to find the velocity of sound in air by resonance column appartus. The difference two resonating lengths is `1.0 m`. Then the velocity of sound in air is

Options

A. ` f m//s`
B. `2 f m//s`
C. `f//2 m//s`
D. `3 f m//s`

Explanation

To find the velocity of sound in air using the resonance column apparatus, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Resonating Lengths**:
- Let \( L_1 \) be the first resonating length and \( L_2 \) be the second resonating length.
- The difference between these two lengths is given as:
\[ L_2 - L_1 = 1.0 \, \text{m} \]

2. **Identifying Nodes and Antinodes**:
- In a resonance column, the first resonance occurs at a length \( L_1 \) where there is a pressure antinode at the open end and a pressure node at the closed end.
- The first resonance length can be expressed as:
\[ L_1 + E = \frac{\lambda}{4} \]
where \( E \) is the end correction and \( \lambda \) is the wavelength.

3. **Second Resonance Length**:
- The second resonance occurs at length \( L_2 \), which can be expressed as:
\[ L_2 + E = \frac{3\lambda}{4} \]

4. **Setting Up the Equations**:
- From the above two equations, we can express them as:
\[ L_1 = \frac{\lambda}{4} - E \quad \text{(1)} \]
\[ L_2 = \frac{3\lambda}{4} - E \quad \text{(2)} \]

5. **Subtracting the Equations**:
- Subtract equation (1) from equation (2):
\[ L_2 - L_1 = \left(\frac{3\lambda}{4} - E\right) - \left(\frac{\lambda}{4} - E\right) \]
- This simplifies to:
\[ L_2 - L_1 = \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2} \]

6. **Using the Given Difference**:
- We know that \( L_2 - L_1 = 1.0 \, \text{m} \):
\[ \frac{\lambda}{2} = 1.0 \, \text{m} \]
- Therefore, solving for \( \lambda \):
\[ \lambda = 2.0 \, \text{m} \]

7. **Calculating the Velocity of Sound**:
- The velocity of sound \( v \) is given by the formula:
\[ v = f \lambda \]
- Substituting the value of \( \lambda \):
\[ v = f \times 2.0 \, \text{m} \]

### Final Answer:
The velocity of sound in air is:
\[ v = 2f \, \text{m/s} \]