Lesson 5 of 10

Distance vs Displacement

路程与位移

Today's question

Run a full lap on a 400 m track: is your displacement 400 m or 0?

Lesson notes

A short written version of Lesson 5. The study sheet has more practice questions.

Short answer

Your displacement is 0, but your distance is 400 m. You end where you started, so your position does not change. The path you ran is 400 m long.

Key ideas

Displacement is the change in position. It is a vector: an arrow from the initial position to the final position. It does not depend on the path.
Distance is the length of the path you actually travel. It is a scalar, just a number, and it depends on the motion.
In 1D, write displacement as a sign plus a number. The sign gives the direction, and the number gives the magnitude.
Displacement = final coordinate − initial coordinate. Never subtract the other way round.
You can't compare distance with displacement directly, because one is a number and the other is an arrow. Compare distance with the magnitude of displacement.
Distance ≥ magnitude of displacement. They are equal only in one-way straight-line motion. If the object turns back, distance is greater.

Step by step

You start at one point on the track, run one full lap, and stop at the same point.
The initial position and the final position are the same point.
Displacement only looks at the change in position, so displacement x = 0.
Distance is the length of the path. One lap of the track is 400 m, so distance s = 400 m.
Check: distance 400 m is greater than the magnitude of displacement, 0. This is allowed, because the path comes back to the start.

Common mistakes

Wrong: saying one lap gives a displacement of 400 m. Right: 400 m is the path length, a distance. Start and end are the same, so displacement is 0.
Wrong: dropping the sign, like writing 2 m for A to C when it is −2 m. Right: use final − initial, and the sign will not get lost.
Wrong: saying distance equals magnitude of displacement in any straight-line motion. Right: this holds only in one-way straight-line motion. With turn-backs, distance is greater.
Wrong: saying distance equals displacement. Right: a number cannot equal an arrow. Say distance equals the magnitude of the displacement.

Practice

Question 1. A ball is thrown straight up from point A, 3 m above the ground. It rises 5 m, then falls back and finally reaches the ground at point C. What is the magnitude of the ball's displacement from A to C? ( )
A. 3 m
B. 5 m
C. 8 m
D. 16 m

Show the answer

A. Displacement only looks at the change in position from start to end. A and C are 3 m apart on a vertical line, so the magnitude is 3 m, pointing down from A to C. For comparison, the distance is 5 + 5 + 3 = 13 m.

Question 2. A person runs 1¾ laps around a horizontal circle of radius R. His ( )
A. Distance and magnitude of displacement are both 3.5πR
B. Distance is 3.5πR; magnitude of displacement is √2R
C. Distance and magnitude of displacement are both R
D. Distance is 0.5πR; magnitude of displacement is √2R

Show the answer

B. Distance: 1¾ = 7/4 laps, each lap is 2πR, so s = 7/4 × 2πR = 3.5πR. A quarter arc is left, so start, centre and end make a right angle. Displacement = √(R² + R²) = √2 R.

Words in English and Chinese

displacement: 位移
distance: 路程
vector: 矢量
scalar: 标量
initial / final position: 初位置 / 末位置
magnitude: 大小
coordinate: 坐标
one-way straight-line motion: 单向直线运动

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