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 given their wavelengths and the number of beats produced. Here’s a step-by-step solution:

### Step 1: Understand the relationship between wavelength, velocity, and frequency
The velocity \( V \) of a wave can be expressed as:
\[ V = f \cdot \lambda \]
where \( f \) is the frequency and \( \lambda \) is the wavelength.

### Step 2: Identify the wavelengths
We are given two wavelengths:
- \( \lambda_1 = 100 \, \text{cm} \)
- \( \lambda_2 = 102 \, \text{cm} \)

### Step 3: Calculate the frequencies of the waves
Since both waves have the same velocity \( V \), we can express their frequencies as:
\[ f_1 = \frac{V}{\lambda_1} = \frac{V}{100} \]
\[ f_2 = \frac{V}{\lambda_2} = \frac{V}{102} \]

### Step 4: Use the beat frequency formula
The beat frequency \( f_b \) is given by the absolute difference of the two frequencies:
\[ f_b = |f_1 - f_2| = \left| \frac{V}{100} - \frac{V}{102} \right| \]
We know that the beat frequency is 12 beats per second:
\[ 12 = \left| \frac{V}{100} - \frac{V}{102} \right| \]

### Step 5: Simplify the equation
We can simplify the equation:
\[ 12 = V \left( \frac{1}{100} - \frac{1}{102} \right) \]
Finding a common denominator (which is 10200):
\[ \frac{1}{100} - \frac{1}{102} = \frac{102 - 100}{10200} = \frac{2}{10200} \]
Thus, we can rewrite the equation as:
\[ 12 = V \cdot \frac{2}{10200} \]

### Step 6: Solve for \( V \)
Now we can solve for \( V \):
\[ V = 12 \cdot \frac{10200}{2} = 12 \cdot 5100 = 61200 \, \text{cm/s} \]

### Step 7: Convert to meters per second
To convert from centimeters per second to meters per second, we divide by 100:
\[ V = \frac{61200}{100} = 612 \, \text{m/s} \]

### Final Answer
The velocity of each wave is:
\[ \boxed{612 \, \text{m/s}} \]