Two waves of wavelength 2m and 2.02 m respectively, moving with the same velocity superpose to produce 2 beats/second. The velocity of the waves is

Options

A. 400 m/s
B. 402 m/s
C. 404 m/s
D. 406 m/s

Explanation

To solve the problem, we need to find the velocity of two waves that produce beats. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the given information
We have two waves with wavelengths:
- \( \lambda_1 = 2 \, \text{m} \)
- \( \lambda_2 = 2.02 \, \text{m} \)

These waves are moving with the same velocity \( v \) and produce 2 beats per second.

### Step 2: Relate frequency to wavelength and velocity
The frequency \( n \) of a wave is given by the formula:
\[ n = \frac{v}{\lambda} \]
where \( v \) is the velocity and \( \lambda \) is the wavelength.

### Step 3: Calculate the frequencies of the two waves
For the first wave:
\[ n_1 = \frac{v}{\lambda_1} = \frac{v}{2} \]

For the second wave:
\[ n_2 = \frac{v}{\lambda_2} = \frac{v}{2.02} \]

### Step 4: Use the information about beats
The number of beats per second is given by the absolute difference in frequencies:
\[ |n_1 - n_2| = 2 \]
Substituting the expressions for \( n_1 \) and \( n_2 \):
\[ \left| \frac{v}{2} - \frac{v}{2.02} \right| = 2 \]

### Step 5: Simplify the equation
To eliminate the absolute value, we can consider:
\[ \frac{v}{2} - \frac{v}{2.02} = 2 \]
Now, find a common denominator:
\[ \frac{v \cdot 2.02 - v \cdot 2}{2 \cdot 2.02} = 2 \]
This simplifies to:
\[ \frac{v(2.02 - 2)}{4.04} = 2 \]
\[ \frac{v(0.02)}{4.04} = 2 \]

### Step 6: Solve for \( v \)
Now, multiply both sides by \( 4.04 \):
\[ v(0.02) = 8.08 \]
Now divide by \( 0.02 \):
\[ v = \frac{8.08}{0.02} = 404 \, \text{m/s} \]

### Conclusion
The velocity of the waves is:
\[ \boxed{404 \, \text{m/s}} \]