A source of sound produces waves of wavelength 60 cm when it is stationary if the speed of sound in air is `320(m)/(s)` and source moves with speed `20(m)/(s)`, the wavelength of sound in the forward direction will be nearest to

Options

A. 56 cm
B. 60 cm
C. 64 cm
D. 68 cm

Explanation

To solve the problem, we need to find the wavelength of sound in the forward direction when the source of sound is moving. We can use the Doppler Effect formula for sound waves to do this.

### Step-by-step Solution:

1. **Identify the given values:**
- Wavelength of sound when stationary, \( \lambda = 60 \, \text{cm} = 0.6 \, \text{m} \)
- Speed of sound in air, \( V = 320 \, \text{m/s} \)
- Speed of the source, \( V_s = 20 \, \text{m/s} \)

2. **Use the Doppler Effect formula:**
The formula for the wavelength of sound when the source is moving towards the observer is given by:
\[ \lambda' = \lambda \left( \frac{V - V_s}{V} \right) \]
where \( \lambda' \) is the new wavelength, \( \lambda \) is the original wavelength, \( V \) is the speed of sound, and \( V_s \) is the speed of the source.

3. **Substitute the values into the formula:**
\[ \lambda' = 0.6 \, \text{m} \left( \frac{320 \, \text{m/s} - 20 \, \text{m/s}}{320 \, \text{m/s}} \right) \]

4. **Calculate the expression inside the parentheses:**
\[ \frac{320 - 20}{320} = \frac{300}{320} = \frac{15}{16} \]

5. **Calculate the new wavelength:**
\[ \lambda' = 0.6 \, \text{m} \times \frac{15}{16} \]
\[ \lambda' = 0.6 \times 0.9375 = 0.5625 \, \text{m} \]

6. **Convert the wavelength back to centimeters:**
\[ \lambda' = 0.5625 \, \text{m} \times 100 = 56.25 \, \text{cm} \]

7. **Round to the nearest whole number:**
The nearest value is \( 56 \, \text{cm} \).

8. **Conclusion:**
The wavelength of sound in the forward direction is approximately \( 56 \, \text{cm} \).

### Final Answer:
The wavelength of sound in the forward direction will be nearest to **56 cm**.

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