A sportsman runs around a circular track of radius \(r\) such that he traverses the path \(A B A B\). The distance travelled and displacement, respectively, are

Explanation:
Let us analyze the motion of the sportsman step-by-step along the circular track:

1. Understanding the Path Structure:
From the given diagram, points A and B are diametrically opposite to each other. The sequence of motion is given as \( ABAB \), meaning the sportsman starts at initial position A and moves continuously as follows:
- From A to B: Covers a semi-circle path length = \( \pi r \)
- From B to A: Covers the next semi-circle path length = \( \pi r \)
- From A to B: Covers another semi-circle path length = \( \pi r \)

2. Calculating Distance Travelled:
Distance is defined as the actual total length of the path covered by the moving body:
\( \text{Total Distance} = \pi r + \pi r + \pi r = 3\pi r \)

3. Calculating Displacement:
Displacement is the shortest straight-line distance separating the initial position and the final position.
- Initial Position: Point A
- Final Position: Point B
Since A and B are opposite ends of the diameter, the straight-line separation between them is equal to the diameter of the track:
\( \text{Displacement} = 2r \)

Hence, Option D is the correct choice.