Introduction:
In this scenario, a particle moves in a circular path of radius R with a constant speed v. The particle starts from point A and returns to it via points B, C, and D. We need to find the average velocity and average acceleration of the particle during this motion.

Understanding the problem:
To solve this problem, we must first understand the concepts of average velocity and average acceleration. Average velocity is defined as the displacement of an object divided by the time taken to travel that displacement. Average acceleration, on the other hand, is the change in velocity divided by the time taken.

Calculating average velocity:
To calculate the average velocity, we need to determine the displacement of the particle and divide it by the total time taken. In this case, since the particle returns to its original position, the displacement is zero. Therefore, the average velocity will also be zero.

Calculating average acceleration:
To calculate the average acceleration, we need to determine the change in velocity and divide it by the total time taken. Since the speed of the particle is constant, there is no change in velocity. Therefore, the average acceleration will also be zero.

Explanation:
1. Average velocity:
- Displacement = 0 (since the particle returns to its original position)
- Total time taken = Time taken to complete one full revolution = Circumference of the circular path / Speed = 2πR / v
- Average velocity = Displacement / Total time taken = 0 / (2πR / v) = 0

2. Average acceleration:
- Change in velocity = 0 (since the speed of the particle is constant)
- Total time taken = Time taken to complete one full revolution = 2πR / v
- Average acceleration = Change in velocity / Total time taken = 0 / (2πR / v) = 0

Conclusion:
In this scenario, the particle moves in a circular path with a constant speed and returns to its original position. As a result, the average velocity and average acceleration of the particle are both zero. This means that the particle maintains a constant speed and does not change its direction during its motion.