In a bicycle the radius of rear wheel is twice the radius of front wheel. If `v_F` and `v_r` are the speeds of top most points of front and rear wheels respectively, then :

Options

A. `v_r = 2 v_F`
B. `v_F = 2 v_r`
C. `v_F = v_r`
D. `v_F gt v_r`

Explanation

To solve the problem, we need to analyze the relationship between the speeds of the topmost points of the front and rear wheels of the bicycle based on their radii.

### Step-by-Step Solution:

1. **Identify the Radii of the Wheels**:
- Let the radius of the front wheel be \( R \).
- Then, the radius of the rear wheel will be \( 2R \) (as given in the problem).

2. **Understand the Relationship Between Linear Speed and Angular Speed**:
- The linear speed \( v \) of a point on the circumference of a wheel is related to its angular speed \( \omega \) and radius \( r \) by the equation:
\[ v = r \cdot \omega \]
- Thus, for the front wheel:
\[ v_F = R \cdot \omega_F \]
- And for the rear wheel:
\[ v_R = 2R \cdot \omega_R \]

3. **Equate the Linear Speeds of the Centers of the Wheels**:
- Since the bicycle is moving forward, the linear speeds of the centers of both wheels must be the same:
\[ v_F = v_R \]

4. **Express the Speeds in Terms of Angular Speeds**:
- From the equations for \( v_F \) and \( v_R \):
\[ R \cdot \omega_F = 2R \cdot \omega_R \]

5. **Simplify the Equation**:
- Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ \omega_F = 2 \cdot \omega_R \]
- This shows that the angular speed of the front wheel is twice that of the rear wheel.

6. **Determine the Speeds of the Topmost Points**:
- The speed of the topmost point of the front wheel \( v_{top,F} \) is:
\[ v_{top,F} = v_F + R \cdot \omega_F \]
- The speed of the topmost point of the rear wheel \( v_{top,R} \) is:
\[ v_{top,R} = v_R + 2R \cdot \omega_R \]
- Since \( v_F = v_R \), we can substitute \( v_F \) for \( v_R \):
\[ v_{top,F} = v_F + R \cdot (2 \cdot \omega_R) = v_F + 2R \cdot \omega_R \]
- Therefore, we find that:
\[ v_{top,F} = v_F + v_R = 2v_R \]
- This means:
\[ v_{top,F} = v_{top,R} \]

7. **Conclusion**:
- The speeds of the topmost points of both wheels are equal, hence:
\[ v_F = v_R \]

### Final Answer:
Both \( v_F \) and \( v_R \) are equal at the topmost points of the front and rear wheels.