Two sound waves of wavelength `1 m` and `1.01 m` in a gas produce `10` beats in `3`s. The velocity of sound in the gas is

Options

A. `(a)330 m//s`
B. `(b)337 m//s`
C. `(c)360 m//s`
D. `(d)300 m//s`

Explanation

To find the velocity of sound in the gas, we will follow these steps:

### Step 1: Identify the given data
We have two wavelengths:
- Wavelength of the first wave, \( \lambda_1 = 1 \, \text{m} \)
- Wavelength of the second wave, \( \lambda_2 = 1.01 \, \text{m} \)

The number of beats produced is 10 in 3 seconds.

### Step 2: Calculate the number of beats per second
The number of beats per second (frequency of beats) can be calculated as:
\[ \text{Beats per second} = \frac{10 \, \text{beats}}{3 \, \text{s}} = \frac{10}{3} \, \text{beats/s} \]

### Step 3: Relate the frequencies of the two waves
Let \( f_1 \) and \( f_2 \) be the frequencies of the first and second waves, respectively. The difference in frequencies gives us the number of beats:
\[ |f_1 - f_2| = \frac{10}{3} \]

### Step 4: Use the formula for frequency
The frequency of a wave is given by the formula:
\[ f = \frac{V}{\lambda} \]
Thus, we can write:
\[ f_1 = \frac{V}{\lambda_1} \quad \text{and} \quad f_2 = \frac{V}{\lambda_2} \]

### Step 5: Set up the equation for the difference in frequencies
Substituting the expressions for \( f_1 \) and \( f_2 \) into the equation for beats gives:
\[ \left|\frac{V}{\lambda_1} - \frac{V}{\lambda_2}\right| = \frac{10}{3} \]

### Step 6: Factor out the common velocity \( V \)
Factoring out \( V \) from the left side:
\[ V \left|\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right| = \frac{10}{3} \]

### Step 7: Substitute the values of wavelengths
Substituting \( \lambda_1 = 1 \, \text{m} \) and \( \lambda_2 = 1.01 \, \text{m} \):
\[ \left|\frac{1}{1} - \frac{1}{1.01}\right| = \left|1 - 0.9901\right| = 0.0099 \]

### Step 8: Solve for \( V \)
Now we can substitute this back into the equation:
\[ V \cdot 0.0099 = \frac{10}{3} \]
\[ V = \frac{10}{3 \cdot 0.0099} \]
Calculating this gives:
\[ V \approx \frac{10}{0.0297} \approx 337.06 \, \text{m/s} \]

### Final Answer
Thus, the velocity of sound in the gas is approximately:
\[ V \approx 337 \, \text{m/s} \]

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