Position, Distance & Displacement

Big Question

You walk from your desk to the classroom door, 6 m away, and then walk straight back to your desk. How far did you walk? Where are you now? How much did your position change? These sound like the same question, but physics gives them three different answers using three different words: position, distance, and displacement. Use the simulations below to figure out how they differ.

Position

symbol: x  ·  vector  ·  unit: m

Where an object is, measured from a chosen reference point (the origin). The sign tells you which side of the origin it is on.

“Where are you right now?”

Displacement

symbol: Δx = xf − xi  ·  vector  ·  unit: m

The straight-line change in position from start to finish, including direction. Only the start and end matter, not the path in between.

“How far, and which way, did you end up from where you started?”

Distance

symbol: d  ·  scalar  ·  unit: m

The total length of the path actually traveled. It has no direction, is never negative, and never goes down while you move.

“How much ground did you cover?”

Part 1 · Walking a Straight Path (1D)

A walker moves along an east–west sidewalk past a house, a tree, and a school. East (→) is the + direction. Click anywhere on the sidewalk to walk there, or use the controls below.

path actually traveled → distance start → finish → displacement origin → walker → position
Position x
0 m
Starting position x₀
0 m
where the trip began
Displacement Δx
0 m
Distance d
0 m
+4 m
x₀ = 0 m
Sample trips:

Position vs. time

The walker moves at a steady 3 m/s, and the clock only runs while it is moving. Peaks and valleys show where it turned around.

Trip log

LegStarts atEnds atLeg displacementLeg distance
Try this: Take any trip, then change the reference point. Which numbers change? Which ones stay exactly the same? Why does that make sense?

Part 2 · Walking Around a Map (2D)

Now the walker can go in any direction on a grid (1 square = 1 m, +x = east, +y = north). Click to drop waypoints; the first click is the starting point. Displacement is always a single straight arrow from start to finish, no matter how twisty the path.

path traveled → distance displacement (start → finish) position vectors (from origin O)
Final position
—
start: —
Displacement
—
Components
—
Δx (east–west), Δy (north–south)
Distance d
—
Sample trips:
Try this: Can you build a path with a distance of 20 m but a displacement of 0 m? What about a displacement of 10 m with the smallest possible distance?

Part 3 · Check Yourself

Read the trip, then find all three quantities. East is the + direction. Use a negative sign for west.

Putting it together

PositionDisplacementDistance
SymbolxΔx = xf − xid
Vector or scalar?vectorvectorscalar
Can it be negative?yesyesno
Depends on the path taken?no (only where you are now)no (only start and end)yes
Depends on where you put the origin?yesnono
Can it decrease while you move?yesyesnever
  • Distance is always greater than or equal to the size of the displacement: d ≥ |Δx|.
  • They are equal only when the object moves in one straight line without turning around.
  • Returning to where you started always gives a displacement of zero, no matter how far you traveled.
  • To combine legs of a trip: add displacements with their signs; add distances as plain sizes.