The velocity of sound in air increases by about. ..... for 1 °C rise in temperature.

Options

A. 10 m/s
B. 0.1 m/s
C. 15.4m/s
D. 0.61 m/s

Explanation

To solve the question regarding the increase in the velocity of sound in air for every 1 °C rise in temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The velocity of sound in air is directly proportional to the temperature. This means that as the temperature increases, the velocity of sound also increases.

2. **Convert Temperature Units**: A rise of 1 °C is equivalent to a rise of 1 K. This is important because the formulas we use often involve Kelvin.

3. **Use the Formula**: The velocity of sound (V) can be expressed using the formula:
\[ V = k \sqrt{T} \]
where \( k \) is a constant and \( T \) is the temperature in Kelvin.

4. **Calculate Initial and Final Velocities**:
- Let’s consider the initial temperature \( T_1 = 273 \) K (which is 0 °C).
- After a 1 °C increase, the new temperature \( T_2 = 274 \) K.

5. **Express the Velocities**:
- Initial velocity at \( T_1 \):
\[ V_1 = k \sqrt{273} \]
- Final velocity at \( T_2 \):
\[ V_2 = k \sqrt{274} \]

6. **Calculate the Increase in Velocity**:
- The increase in velocity \( \Delta V \) can be found by subtracting the initial velocity from the final velocity:
\[ \Delta V = V_2 - V_1 = k \sqrt{274} - k \sqrt{273} \]

7. **Simplify the Expression**:
- Factor out \( k \):
\[ \Delta V = k (\sqrt{274} - \sqrt{273}) \]
- To approximate \( \sqrt{274} - \sqrt{273} \), we can use the binomial expansion:
\[ \sqrt{1 + x} \approx 1 + \frac{x}{2} \text{ for small } x \]
- Here, \( x = \frac{1}{273} \):
\[ \sqrt{274} \approx \sqrt{273} \left(1 + \frac{1}{2 \times 273}\right) \]

8. **Calculate the Approximate Increase**:
- Using the approximation:
\[ \Delta V \approx k \left(\sqrt{273} \cdot \frac{1}{2 \times 273}\right) = \frac{k}{2 \sqrt{273}} \]
- Given that the approximate value of \( V \) at 0 °C is around 330 m/s, we can substitute \( k \) with this value.

9. **Final Calculation**:
- Plugging in the values:
\[ \Delta V \approx \frac{330}{2 \times 273} \approx 0.61 \text{ m/s} \]

### Conclusion:
The velocity of sound in air increases by approximately **0.61 m/s** for every 1 °C rise in temperature.

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