Q.Three girls skating on a circular ice ground of radius $200\ \text{m}$ start from a point $P$ on the edge of the ground and reach a point $Q$ that is diametrically opposite to $P$, each following a different path across the ground. The first girl follows a curved path that bulges out to the left, the second girl travels along the straight diameter directly from $P$ to $Q$, and the third girl follows a wavy (zig-zag) path that bulges out to the right, as shown below.

Displacement depends only on the endpoints. Since $P$ and $Q$ are diametrically opposite, $|\text{displacement}| = 2R = 2 \times 200 = 400\ \text{m}$ for every girl. Path length equals this only when the route is a straight line — the middle girl on the diame …

Concept

The displacement is the straight-line vector from the initial position to the final position; its magnitude depends only on the endpoints, not on the route taken. The path length depends on the actual route travelled and is always greater than or equal to the displacement magnitude.

Given

• Radius of the ground $R = 200\ \text{m}$.
• $P$ and $Q$ are diametrically opposite, so the segment $PQ$ is a diameter.

Displacement magnitude

Because all three girls begin at $P$ and end at $Q$,

for each of them, regardless of the curved route followed.

Path length equal to displacement …

Concept: Coordinates Plus the Shortest-Path Theorem

Method: Place $P,Q$ on Coordinates, Compute Displacement by the Distance Formula, Then Invoke the Shortest-Path Theorem for the Path-Length Comparison

Rather than reasoning purely verbally ("displacement only depends on endpoints"), this method sets up explicit coordinates for $P$ and $Q$, computes the displacement vector's magnitude directly by the distance formula, and then settles the path-length question by citing the general mathematical theorem that a straight line is the unique shortest curve joining two points.

Step 1 -- Place the circle on coordinates

Let the centre of the circular ground be the origin, with radius $R=200\ \text{m}$. Since $P$ and $Q$ are diametrically opposite, place them at the two ends of a diameter along the $x$-axis:

Step 2 -- Displacement vector, by the distance formula

The displacement for any of the three girls is the same vector $\overrightarrow{PQ}$, since all three start at $P$ and end at $Q$ -- regardless of the route taken between:

Step 3 -- Invoke the shortest-path theorem for the path-length question

A standard theorem of Euclidean geometry states: among all curves connecting two given points, the straight-line segment joining them is the unique curve of minimum length; every other curve connecting the same two points is strictly longer.

Applying this directly:

• The middle girl's path is the straight segment $PQ$ itself, so her path length equals exactly $|PQ| = 400\ \text{m}$ -- matching the displacement magnitude exactly. …