P,Q and R are three particle of a medium which lie on the x-axis. A sin wave of wavelength `lamda` is travelling through the medium in the x-direction. P and Q always have the same speed. Whilem P and R always have the same velocity. The minimum distance between :-

Options

A. P and Q is `(lamda)/(2)`
B. P and Q is `lamda`
C. P and R is `(lamda)/(2)`
D. P and R is `lamda`

Answer

To solve the problem, we need to analyze the motion of the three particles P, Q, and R in relation to the sine wave traveling through the medium.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have three particles P, Q, and R on the x-axis.
- A sine wave of wavelength \( \lambda \) is traveling in the positive x-direction.
- P and Q have the same speed, while P and R have the same velocity.

2. **Speed vs. Velocity**:
- Speed is a scalar quantity and does not have a direction, while velocity is a vector quantity that includes direction.
- Since P and Q have the same speed, they can be moving in different directions but at the same rate.
- Since P and R have the same velocity, they must be moving in the same direction and at the same rate.

3. **Determining the Speeds**:
- Let's denote the speed of P and Q as \( v \).
- Since they are moving with the same speed, we can say \( v_P = v_Q = v \).
- For P and R, since they have the same velocity, we can denote their velocity as \( v_P = v_R \).

4. **Distance Between Particles**:
- The distance between P and Q can be determined based on their speeds and the time they have been moving.
- Since P and Q have the same speed, the distance between them will depend on their initial positions and the time they have been moving.
- The distance between P and R will be determined by their velocities.

5. **Minimum Distance Calculation**:
- Given that the wavelength of the sine wave is \( \lambda \), and since P and R have the same velocity, the distance between them can be considered as \( \lambda \) when they are in phase with the wave.
- Therefore, the minimum distance between P and R is equal to the wavelength \( \lambda \).

6. **Conclusion**:
- The minimum distance between the particles P, Q, and R is determined by their respective speeds and velocities in relation to the sine wave.
- The minimum distance between P and R is \( \lambda \).

### Final Answer:
The minimum distance between particles P, Q, and R is \( \lambda \).

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