At room temperature `(27^(@)C)` the velocity of sound in air is 330 m/s. the increase in velocity of sound when temperature is increased by `1^(@)C` is

Options

A. 2 m/s
B. 1 m/s
C. 10 m/s
D. 0.55 m/s

Answer

To solve the problem of finding the increase in the velocity of sound when the temperature is increased by \(1^\circ C\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between temperature and sound velocity**:
The velocity of sound in air is directly proportional to the square root of the absolute temperature (in Kelvin). The formula can be expressed as:
\[ v \propto \sqrt{T} \]
where \(v\) is the velocity of sound and \(T\) is the temperature in Kelvin.

2. **Convert the initial temperature to Kelvin**:
The initial temperature is given as \(27^\circ C\). To convert this to Kelvin:
\[ T_1 = 27 + 273 = 300 \, K \]

3. **Calculate the new temperature after increasing by \(1^\circ C\)**:
The new temperature will be:
\[ T_2 = 27 + 1 + 273 = 301 \, K \]

4. **Use the proportional relationship to find the new velocity**:
The initial velocity of sound at \(T_1\) is given as \(v_1 = 330 \, m/s\). We can set up the ratio of the velocities:
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \]
Substituting the values:
\[ \frac{v_2}{330} = \sqrt{\frac{301}{300}} \]

5. **Calculate \(v_2\)**:
First, calculate \(\sqrt{\frac{301}{300}}\):
\[ \sqrt{\frac{301}{300}} \approx 1.0033 \]
Now, substituting back to find \(v_2\):
\[ v_2 = 330 \times 1.0033 \approx 331.09 \, m/s \]

6. **Find the increase in velocity**:
The increase in velocity \(\Delta v\) is given by:
\[ \Delta v = v_2 - v_1 = 331.09 - 330 \approx 1.09 \, m/s \]

### Final Answer:
The increase in velocity of sound when the temperature is increased by \(1^\circ C\) is approximately \(1.09 \, m/s\).

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