Distance and Displacement

The video lesson for this topic is coming soon.

Essential Question

How far did you go, and where did you end up?

Walk around the block and come home, and you’ve gone a long way but ended up right where you started. By the end of this lesson, you should be able to find the distance and the displacement for a trip, and tell the difference between a scalar and a vector.

Position

Where an object is
Measured from a reference point
Unit: meters (m)

Position tells you where an object is. It is always measured from a reference point, a spot you choose to call zero. Here the tree is the reference point, and the person is standing 6 m to the right of it, so their position is +6 m. Position is measured in meters (m).

Choosing Directions

Pick a reference point (zero)
One direction is positive (+)
The other is negative (−)

A number line from −4 to +4 meters with a house at 0. An arrow pointing right is labeled + east and an arrow pointing left is labeled − west. A blue dot at +3 m and a red dot at −3 m.

Once you choose a reference point, you also choose which direction counts as positive. Here east (right) is positive and west (left) is negative. A position of +3 m means 3 m east of the house, and −3 m means 3 m west of it. The sign is how a single number can show direction.

Distance

Total length of the path
Never negative
A scalar: size only

Distance is the total length of the path an object travels. If the winding route from home to school is 900 m long, you traveled a distance of 900 m, no matter which direction the path turned. Distance can never be negative, and it has no direction, so it is a scalar: a quantity with size only.

Displacement

Change in position
Straight line from start to end
A vector: size + direction

The same home and school. A gray dashed winding path labeled path 900 m, and a straight red arrow from home to school labeled 600 m east.

Displacement is the straight-line change in position from where you started to where you ended, including direction. The path to school is 900 m long, but school is only 600 m east of home, so the displacement is 600 m east. Because displacement has both a size and a direction, it is a vector.

Finding Displacement

Δx = x_f − x_i

Δx = displacement, in meters (m)
x_f = final position, in meters (m)
x_i = initial position, in meters (m)

The Greek letter Δ (delta) means “change in,” so Δx means change in position. Always subtract final minus initial. A positive answer means the displacement is in the positive direction, and a negative answer means it is in the negative direction. You will often see displacement written simply as d.

Scalars vs. Vectors

Two cards. Left, cyan: Scalar, size only, with examples distance, speed, time, and mass, and the sample value 5 m. Right, pink: Vector, size + direction, with examples displacement, velocity, and force, and the sample value 5 m east.

Every quantity in physics is either a scalar or a vector. A scalar has a size only, like a distance of 5 m, a time of 60 s, or a mass of 2 kg. A vector has a size and a direction, like a displacement of 5 m east. Vectors are drawn as arrows, where the length shows the size and the arrowhead shows the direction.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Hallway Walk

A student walks 12 m east down a hallway, then turns around and walks 5 m west to a drinking fountain. Find her distance and her displacement.

A blue arrow pointing right labeled 12 m east, and below it a red arrow pointing left from the end of the first arrow, labeled 5 m west. The start is marked on the left.

Distance: add up every part of the path: 12 m + 5 m = 17 m
Choose directions: call east positive (+) and west negative (−)
Displacement: Δx = (+12 m) + (−5 m) = +7 m
Answer: 7 m east. The positive sign means east.

Distance and displacement are different because she turned around. They are only equal when an object moves in one direction the whole time.

Example 2: Using Positions

A soccer ball starts at position +3.0 m and rolls to position −5.0 m. East is positive. What is the ball’s displacement?

A number line from −6 to +4 meters. A ball labeled start sits at +3 m and a dashed circle labeled end sits at −5 m. A note says + = east.

Knowns: x_i = +3.0 m, x_f = −5.0 m
Equation: Δx = x_f − x_i
Substitute: Δx = (−5.0 m) − (+3.0 m) = −8.0 m
Answer: Δx = 8.0 m west. The negative sign means west.

Example 3: Elevator Ride

An elevator starts 12 m above the ground. It rises to 33 m, then goes down to 6 m. Up is positive. Find the elevator’s distance and displacement.

A vertical scale showing heights of 0, 6, 12, and 33 meters above the ground. A small elevator car sits at 12 m, labeled start. Dashed outlines of the car at 33 m and 6 m are labeled top and end.

Distance: up: 33 m − 12 m = 21 m. Down: 33 m − 6 m = 27 m. Total: 21 m + 27 m = 48 m
Displacement equation: Δx = x_f − x_i
Substitute: Δx = 6 m − 12 m = −6 m
Answer: Δx = 6 m down. The negative sign means down.

The trip to the top doesn’t change the displacement at all. Only the starting and ending positions matter.

Example 4: Scalar or Vector?

Which of these are vectors? A) 25 m north B) 60 s C) 15 m/s west D) 9 m

A) 25 m north: has a size and a direction, so it is a vector (a displacement)
B) 60 s: time has no direction, so it is a scalar
C) 15 m/s west: has a size and a direction, so it is a vector (a velocity)
D) 9 m: a size with no direction, so it is a scalar (a distance)
Answer: A and C are vectors

Big Ideas

Position: where you are, measured from zero
Distance: total path length (scalar)
Displacement: Δx = x_f − x_i (vector)
Vectors have size and direction

Back to the essential question: “how far you went” is your distance, the total length of your path. “Where you ended up” is your displacement, the straight-line change in position from start to finish, with a direction. Next up, we’ll use distance and displacement to describe how fast something moves with speed and velocity.