Assertion: A particle is rotating in a circle with constant speed as shown. Between point `A` and `B` , ratio of average acceleration and average velocity `v` can be stopped in a minimum distance `d` . If the same car, moving with same speed `v` takes a circular turn, then minimum safe radius can be `2d` .
Reasion: `d=(v_(2))/(2mug)` and minimum safe radius `=(v_(2))/(mug)`

Options

A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, true but the Reason is false.
D. If Assertion is false but the Reason is true.

Answer

To solve the problem, we need to analyze the assertion and reason provided, and then derive the necessary relationships between average acceleration, average velocity, and the radius of the circular path.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that a particle moving in a circle with constant speed can stop in a minimum distance \(d\) when moving between points A and B. This implies that the average acceleration and average velocity have a specific relationship.

2. **Average Velocity and Average Acceleration**:
- The average velocity \(v_{avg}\) can be defined as the total displacement divided by the total time taken. For a circular motion, if the particle moves from point A to point B, the average velocity can be expressed as:
\[ v_{avg} = \frac{\Delta s}{\Delta t} \]
- The average acceleration \(a_{avg}\) can be defined as the change in velocity divided by the time taken:
\[ a_{avg} = \frac{\Delta v}{\Delta t} \]

3. **Using the Stopping Distance**:
- The stopping distance \(d\) can be derived from the equation of motion:
\[ d = \frac{v^2}{2a} \]
- Rearranging gives us:
\[ a = \frac{v^2}{2d} \]

4. **Minimum Safe Radius**:
- The reason states that the minimum safe radius \(R\) for the car taking a circular turn at speed \(v\) is given by:
\[ R = \frac{v^2}{\mu g} \]
- Here, \(\mu\) is the coefficient of friction and \(g\) is the acceleration due to gravity.

5. **Connecting the Two Concepts**:
- From the stopping distance equation, we have:
\[ d = \frac{v^2}{2\mu g} \]
- Therefore, if we double the stopping distance, we can express the minimum safe radius as:
\[ R = 2d = \frac{v^2}{\mu g} \]

6. **Conclusion**:
- Both the assertion and the reason are correct. The assertion correctly describes the relationship between average acceleration and average velocity in terms of stopping distance, and the reason correctly describes the minimum safe radius for circular motion.

### Final Answer:
- Both the assertion and the reason are correct, but the reason is not the correct explanation for the assertion.