A sportsman runs around a circular track of radius $r$ such that he traverses the path ABAB, where A and B are the ends of a diameter. The distance travelled and displacement, respectively, are:
- A$2r,\ 3\pi r$
- B$3\pi r,\ \pi r$
- C$\pi r,\ 3r$
- D$3\pi r,\ 2r$
Answer and solution
Answer: Option D: $3\pi r,\ 2r$
Concept. Distance is the length of the path actually covered, so it keeps adding up along the track. Displacement is the straight-line vector from the starting point to the final point, so it depends only on where the motion starts and ends.
Step 1. Break the path into pieces. A and B are the ends of a diameter, so going from one to the other along the track is half a lap. The path ABAB has three such pieces: A to B, B to A, A to B.
Step 2. Distance. Each half lap is half the circumference: $$\text{half lap}=\frac{2\pi r}{2}=\pi r\qquad\Rightarrow\qquad \text{distance}=3\times\pi r=3\pi r$$
Step 3. Displacement. He starts at A and finishes at B. The straight line from A to B is the diameter: $$|\text{displacement}|=AB=2r$$
Answer. $3\pi r,\ 2r$ (Option D). The full lap from A back to A adds $2\pi r$ to the distance but nothing to the displacement.
Point to note. Distance adds every half lap ($\pi r$); displacement only compares start and end.
