To determine the distance traveled and displacement of a sportsman running around a circular track, we begin by analyzing the path described as $ A \rightarrow B \rightarrow A \rightarrow B $.

Displacement:

The displacement is the straight line distance from the initial point to the final point. In this case, the sportsman completes one full circle (from $ A $ back to $ A $) and then continues to point $ B $ (which is opposite point $ A $ on the circle). Therefore, the displacement is the diameter of the circle:

$ \text{Displacement} = 2r $

Distance:

The distance is the total path length traveled by the sportsman. Starting at $ A $, he completes one full circle back to $ A $, which has a circumference of:

$ 2 \pi r $

He then runs half the circle from $ A $ to $ B $, which is another:

$ \frac{1}{2} \times 2 \pi r = \pi r $

Thus, the total distance is the circumference of the full circle plus half of another circle:

$ \text{Distance} = 2 \pi r + \pi r = 3 \pi r $

In summary, the distance traveled by the sportsman is $ 3\pi r $ and the displacement is $ 2r $.