The speed of sound wave in a gas, in which two waves of wavelengths 1.0m and 1.02m produce 6 beats per second is

Options

A. 300m/s
B. 306m/s
C. 380m/s
D. 410m/s

Answer

To solve the problem, we need to find the speed of sound in a gas given the wavelengths of two sound waves and the number of beats produced. Here’s a step-by-step solution:

### Step 1: Understand the relationship between wavelength, frequency, and speed
The speed of a wave (v) is given by the formula:
\[ v = f \cdot \lambda \]
where \( f \) is the frequency and \( \lambda \) is the wavelength.

### Step 2: Identify the given values
We have:
- Wavelength 1 (\( \lambda_1 \)) = 1.0 m
- Wavelength 2 (\( \lambda_2 \)) = 1.02 m
- Number of beats per second = 6 beats/s

### Step 3: Relate the frequencies of the two waves
The frequency of a wave can be expressed as:
\[ f_1 = \frac{v}{\lambda_1} \]
\[ f_2 = \frac{v}{\lambda_2} \]

The difference in frequencies (which produces beats) is given by:
\[ |f_1 - f_2| = 6 \]

### Step 4: Substitute the expressions for frequencies
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \left| \frac{v}{\lambda_1} - \frac{v}{\lambda_2} \right| = 6 \]

### Step 5: Substitute the values of wavelengths
Substituting \( \lambda_1 = 1.0 \) m and \( \lambda_2 = 1.02 \) m:
\[ \left| \frac{v}{1.0} - \frac{v}{1.02} \right| = 6 \]

### Step 6: Simplify the equation
This can be simplified to:
\[ \left| v - \frac{v \cdot 1.0}{1.02} \right| = 6 \]
\[ \left| v - \frac{v}{1.02} \right| = 6 \]

### Step 7: Factor out v
Factoring out \( v \):
\[ v \left( 1 - \frac{1}{1.02} \right) = 6 \]
\[ v \left( \frac{1.02 - 1}{1.02} \right) = 6 \]
\[ v \left( \frac{0.02}{1.02} \right) = 6 \]

### Step 8: Solve for v
Now, rearranging gives:
\[ v = 6 \cdot \frac{1.02}{0.02} \]
\[ v = 6 \cdot 51 = 306 \text{ m/s} \]

### Conclusion
The speed of sound in the gas is:
\[ \boxed{306 \text{ m/s}} \]

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