NCERT Solutions

We need to calculate the displacement magnitude for three skaters and identify whose path length matches this displacement magnitude.

Concept: Displacement

Displacement is defined as the shortest straight-line distance between the initial and final positions of an object. Its magnitude is independent of the actual path taken. The distance is the total length of the path traveled. For a circular ground, the distance between two diametrically opposite points is equal to the diameter of the circle:

$$d = 2r$$

The magnitude of displacement equals the path length only if the motion is in a straight line without changing direction.

Given:

• Radius of the circular ice ground, $r = 200\,\text{m}$
• Initial position: Point $P$
• Final position: Point $Q$ (diametrically opposite to $P$)

Solving:

1. The displacement vector for each girl is the vector $\vec{PQ}$ pointing from the starting point $P$ to the finishing point $Q$.
2. Since $P$ and $Q$ are diametrically opposite, the magnitude of this displacement is equal to the diameter of the circle.

$$\begin{aligned} |\vec{s}| &= 2r \\ &= 2 \times 200\,\text{m} \\ &= 400\,\text{m} \end{aligned}$$

3. All three girls (A, B, and C) start at $P$ and end at $Q$. Therefore, the magnitude of the displacement is the same for all three girls, which is $400\,\text{m}$.
4. By observing the paths, Girl B travels along the straight line $PQ$, which is the diameter. Girls A and C follow curved paths.
5. For Girl B, the actual length of the path skated is equal to the magnitude of the displacement because she moves along a straight line.

Answer

The magnitude of the displacement for each girl is:

$$400\,\text{m}$$

This is equal to the actual path length for Girl B.