A sine wave has an amplitude A and wavelength `lambda`. Let V be the wave velocity and v be the maximum velocity of a particle in the medium. Then

Options

A. V cannot to equal to v
B. `V=v, "if" A = lambda//2pi`
C. `V=v , "if" A =2pilambda`
D. `V=v , "if" lambda=A//pi`

Answer

To solve the problem, we will analyze the relationships between wave properties such as amplitude, wavelength, wave velocity, and maximum particle velocity.

### Step-by-Step Solution:

1. **Understanding the Variables**:
- Let \( A \) be the amplitude of the sine wave.
- Let \( \lambda \) be the wavelength of the wave.
- Let \( V \) be the wave velocity.
- Let \( v \) be the maximum velocity of a particle in the medium.

2. **Maximum Velocity of a Particle**:
- The maximum velocity \( v \) of a particle in the medium is given by the formula:
\[ v = A \cdot \omega \]
where \( \omega \) is the angular frequency of the wave.

3. **Angular Frequency**:
- The angular frequency \( \omega \) can be expressed in terms of the frequency \( f \):
\[ \omega = 2\pi f \]
Thus, substituting this into the equation for maximum velocity, we get:
\[ v = A \cdot (2\pi f) \]
This can be labeled as Equation (1).

4. **Wave Velocity**:
- The wave velocity \( V \) can also be expressed in terms of frequency and wavelength:
\[ V = f \cdot \lambda \]
This can be labeled as Equation (2).

5. **Equating Wave Velocity and Maximum Velocity**:
- According to the problem, we need to find the condition under which the wave velocity \( V \) equals the maximum particle velocity \( v \). Therefore, we set:
\[ V = v \]
Substituting the expressions from Equations (1) and (2):
\[ f \cdot \lambda = A \cdot (2\pi f) \]

6. **Simplifying the Equation**:
- We can cancel \( f \) from both sides (assuming \( f \neq 0 \)):
\[ \lambda = A \cdot (2\pi) \]
Rearranging gives:
\[ A = \frac{\lambda}{2\pi} \]

7. **Conclusion**:
- We have derived that the amplitude \( A \) is related to the wavelength \( \lambda \) by the equation:
\[ A = \frac{\lambda}{2\pi} \]
This means that if \( A = \frac{\lambda}{2\pi} \), then the wave velocity \( V \) will equal the maximum velocity \( v \).

### Final Answer:
The correct condition is \( v = V \) if \( A = \frac{\lambda}{2\pi} \).