A ring performs pure rolling on a horizontal fixed surface with its centre moving with speed v.

Options

A. Maximum possible speed that any particle of ring may have is `v//sqrt(2)`
B. Minimum possible speed that any particle of ring may have is `v//sqrt(2)`
C. particles on the ring may have speed anywhere between 0 and 2u.
D. None of these

Explanation

To solve the problem of a ring performing pure rolling on a horizontal surface with its center moving at speed \( v \), we need to analyze the velocities of different points on the ring.

### Step-by-Step Solution:

1. **Understanding Pure Rolling**:
- In pure rolling motion, the point of contact between the ring and the surface is momentarily at rest. This means that the velocity of the center of the ring, \( v \), is equal to the linear velocity of the ring's center.

2. **Identifying the Points on the Ring**:
- Consider a ring of radius \( r \). The center of the ring moves with speed \( v \).
- The ring has points at the top, bottom, and sides. We will analyze the velocities at these points.

3. **Velocity at the Point of Contact**:
- The point of contact with the ground has a velocity of \( 0 \) because it is at rest during pure rolling.

4. **Velocity at the Topmost Point**:
- The topmost point of the ring moves forward with the speed of the center plus the speed due to rotation.
- The linear speed due to rotation at the topmost point is \( \omega r \), where \( \omega \) is the angular velocity.
- For pure rolling, we have the relationship \( v = \omega r \). Thus, the speed at the topmost point is:
\[ v_{\text{top}} = v + \omega r = v + v = 2v \]

5. **Velocity at the Bottommost Point**:
- The bottommost point of the ring is moving backward with the speed of the center minus the speed due to rotation.
- Hence, the speed at the bottommost point is:
\[ v_{\text{bottom}} = v - \omega r = v - v = 0 \]

6. **Conclusion on Velocities**:
- The velocities of the particles on the ring range from \( 0 \) (at the bottommost point) to \( 2v \) (at the topmost point).
- Therefore, the particles on the ring can have speeds anywhere between \( 0 \) to \( 2v \).

7. **Answering the Options**:
- Maximum possible speed at any particle of the ring is \( 2v \).
- Minimum possible speed at any particle is \( 0 \).
- The particles on the ring may have speeds anywhere between \( 0 \) to \( 2v \).

### Final Answer:
- The correct option is: "Particle on the ring may have speed anywhere between \( 0 \) to \( 2v \)."