Path vs Shortcut

Why it matters

Your morning walk: 2 km of steps, ending back at your own gate. The odometer says 2 km. The crow flying home says ZERO. Both are right — they measure different things.

The idea in plain words

Odometer vs arrow. Tap the terms.

distance ≥ |displacement|

A body goes 12 m east, then 5 m north. Distance and displacement?

17 m walked, 13 m displaced

• the odometer: distance — total path length — scalar, never decreases, never negative
• the crow's flight: displacement — straight line from start to finish, with direction — the vector shortcut
• the special case: equality — they match only for straight-line, one-way motion
• distance = 12 + 5 = 17 m
• displacement = √(144 + 25) = √169

Predict first

You walk one full lap of a circular park of radius 70 m. Distance and displacement?

Distance counts every step of the path. Displacement is the straight arrow start→end — and yours has zero length.

• distance = 2πr ≈ 440 m, displacement = 0 — correct
• both ≈ 440 m
• both zero — you're back where you started

What you do

Walk the grid with the arrows: pile up 400 m of distance, then bring the displacement back to exactly zero.

Check yourself

Displacement can be…

Any round trip: kilometres of distance, zero displacement. The shortcut can never exceed the path.

Half a lap around a circle of radius r gives displacement…

Start and end sit at opposite ends of a diameter: straight-line separation 2r. (Distance = πr.)

When are distance and |displacement| equal?

March straight and the path IS the shortcut. One U-turn and they part company.

• zero even when distance is large — correct
• greater than distance
• negative distance
• 2r (the diameter) — correct
• πr
• zero
• straight-line motion without turning back — correct
• always
• never