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. `330 m//s`
B. `337 m//s`
C. `360 m//s`
D. `300 m//s`

Explanation

To solve the problem, we need to determine the velocity of sound in the gas based on the given wavelengths and the number of beats produced. Here is a step-by-step solution:

### Step 1: Identify Given Values
- Wavelength of the first wave, \( \lambda_1 = 1 \, \text{m} \)
- Wavelength of the second wave, \( \lambda_2 = 1.01 \, \text{m} \)
- Number of beats produced in 3 seconds = 10

### Step 2: Calculate Beats Per Second
To find the frequency of beats per second, we divide the total number of beats by the time in seconds:
\[ \text{Beats per second} = \frac{10 \, \text{beats}}{3 \, \text{s}} = \frac{10}{3} \, \text{Hz} \approx 3.33 \, \text{Hz} \]

### Step 3: Relate Frequency and Wavelength to Velocity
The frequency \( f \) of a wave is related to its wavelength \( \lambda \) and velocity \( v \) by the formula:
\[ f = \frac{v}{\lambda} \]
Thus, for the two waves, we have:
\[ f_1 = \frac{v}{\lambda_1} \quad \text{and} \quad f_2 = \frac{v}{\lambda_2} \]

### Step 4: Calculate the Difference in Frequencies
The number of beats per second is given by the difference in frequencies:
\[ |f_1 - f_2| = \frac{v}{\lambda_1} - \frac{v}{\lambda_2} = \frac{10}{3} \]

### Step 5: Substitute the Wavelengths
Substituting \( \lambda_1 \) and \( \lambda_2 \):
\[ \frac{v}{1} - \frac{v}{1.01} = \frac{10}{3} \]

### Step 6: Factor Out \( v \)
Factoring out \( v \):
\[ v \left( 1 - \frac{1}{1.01} \right) = \frac{10}{3} \]

### Step 7: Simplify the Expression
Calculating \( 1 - \frac{1}{1.01} \):
\[ 1 - \frac{1}{1.01} = \frac{1.01 - 1}{1.01} = \frac{0.01}{1.01} \]
Thus, we can rewrite the equation:
\[ v \left( \frac{0.01}{1.01} \right) = \frac{10}{3} \]

### Step 8: Solve for \( v \)
Now, solving for \( v \):
\[ v = \frac{10}{3} \cdot \frac{1.01}{0.01} \]
\[ v = \frac{10.1}{0.01} \cdot \frac{1}{3} = \frac{1010}{3} \approx 336.67 \, \text{m/s} \]

### Step 9: Final Result
Rounding to the nearest whole number, the velocity of sound in the gas is approximately:
\[ v \approx 337 \, \text{m/s} \]

### Answer
The velocity of sound in the gas is approximately **337 m/s**.

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