two waves of wavelength 100 cm and 102 cm produce 12 beats per second having same velocity. the velocity of each wave is

Options

A. 320 m/s
B. 612 m/s
C. 306 m/s
D. 220 m/s

Answer

To solve the problem, we need to find the velocity of two waves with given wavelengths that produce a certain number of beats per second. Here’s a step-by-step solution:

### Step 1: Understand the relationship between frequency, wavelength, and velocity
The frequency \( f \) of a wave is related to its velocity \( V \) and wavelength \( \lambda \) by the formula:
\[ f = \frac{V}{\lambda} \]
where \( V \) is the velocity of the wave and \( \lambda \) is the wavelength.

### Step 2: Identify the given values
We are given:
- Wavelength of the first wave, \( \lambda_1 = 100 \) cm
- Wavelength of the second wave, \( \lambda_2 = 102 \) cm
- Number of beats per second, \( \text{beats} = 12 \) beats/s

### Step 3: Relate the beat frequency to the difference in frequencies of the two waves
The beat frequency is equal to the difference in frequencies of the two waves:
\[ |f_1 - f_2| = 12 \text{ beats/s} \]

### Step 4: Express the frequencies in terms of velocity and wavelength
Using the formula for frequency, we can express the frequencies of the two waves:
\[ f_1 = \frac{V}{\lambda_1} = \frac{V}{100} \]
\[ f_2 = \frac{V}{\lambda_2} = \frac{V}{102} \]

### Step 5: Set up the equation for the beat frequency
Substituting the expressions for \( f_1 \) and \( f_2 \) into the beat frequency equation:
\[ \left| \frac{V}{100} - \frac{V}{102} \right| = 12 \]

### Step 6: Simplify the equation
To simplify:
\[ \frac{V}{100} - \frac{V}{102} = 12 \]
Finding a common denominator (which is \( 10200 \)):
\[ \frac{102V - 100V}{10200} = 12 \]
This simplifies to:
\[ \frac{2V}{10200} = 12 \]

### Step 7: Solve for \( V \)
Multiplying both sides by \( 10200 \):
\[ 2V = 12 \times 10200 \]
\[ 2V = 122400 \]
Dividing by 2:
\[ V = 61200 \text{ cm/s} \]

### Step 8: Convert to meters per second
Since the problem asks for the velocity in meters per second:
\[ V = \frac{61200}{100} = 612 \text{ m/s} \]

### Final Answer
The velocity of each wave is \( 612 \text{ m/s} \).

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