A wheel of radius R rolls on the ground with a uniform velocity v. The velocity of topmost point relative to the bottommost point is

Options

A. v
B. 2v
C. v/2
D. zero

Explanation

To find the velocity of the topmost point of a wheel rolling on the ground relative to the bottommost point, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Points on the Wheel:**
- Let the topmost point of the wheel be denoted as point P and the bottommost point (the point of contact with the ground) be denoted as point O.

2. **Understand the Motion of the Wheel:**
- The wheel rolls on the ground with a uniform velocity \( v \). In pure rolling motion, the velocity of the bottommost point O (the point in contact with the ground) is \( 0 \, \text{m/s} \).

3. **Determine the Angular Velocity:**
- The wheel is rolling without slipping, which means it has both translational and rotational motion. The relationship between the linear velocity \( v \) of the center of mass and the angular velocity \( \omega \) is given by:
\[ v = \omega R \]
- Here, \( R \) is the radius of the wheel.

4. **Calculate the Velocity of the Topmost Point:**
- The distance from the center of the wheel to the topmost point P is \( R \). The velocity of point P can be calculated using the formula:
\[ V_P = \omega \times R + v \]
- Since the wheel is rolling, the velocity of point P relative to the center of the wheel is \( \omega R \) upwards, and the center of mass is moving with velocity \( v \) to the right. Thus:
\[ V_P = \omega R + v = v + v = 2v \]

5. **Calculate the Relative Velocity:**
- To find the velocity of the topmost point P relative to the bottommost point O, we use the formula:
\[ V_{PO} = V_P - V_O \]
- Since \( V_O = 0 \, \text{m/s} \) (the bottommost point), we have:
\[ V_{PO} = V_P - 0 = V_P \]
- Therefore:
\[ V_{PO} = 2v \]

6. **Conclusion:**
- The velocity of the topmost point relative to the bottommost point is \( 2v \).

### Final Answer:
The velocity of the topmost point relative to the bottommost point is \( 2v \).