Question

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

Answer 1

3πr, 2r

Answer 2

The sportsman starts at point A and traverses the path ABAB. Based on the figure, A and B are the endpoints of a diameter of the circular track, and the path from A to B is a semi-circle. We interpret the path ABAB as starting at A, going to B, then returning to A, and finally going to B again. Thus, the path is A $\to$ B $\to$ A $\to$ B.

The path from A to B is a semi-circle of radius r. The length of this semi-circle is half the circumference of the circle, which is $\frac{1}{2} \times 2\pi r = \pi r$. The path from B to A along the semi-circle also has a length of $\pi r$.

The total distance travelled is the sum of the lengths of the individual paths: Distance = (Distance from A to B) + (Distance from B to A) + (Distance from A to B) Distance = $\pi r + \pi r + \pi r = 3\pi r$.

The displacement is the shortest distance between the initial position and the final position. The initial position is A. The path is A $\to$ B $\to$ A $\to$ B. After traversing A $\to$ B, the position is B. After traversing B $\to$ A, the position is A. After traversing A $\to$ B, the final position is B.

The initial position is A and the final position is B. Since A and B are the endpoints of a diameter, the straight line distance between A and B is the diameter, which is $2r$. The displacement is the vector from A to B, and its magnitude is $2r$.

Therefore, the distance travelled is $3\pi r$ and the displacement is $2r$.