Position, distance and displacement

Run one full lap of a 400 m track and you have covered 400 m of distance, yet your displacement is zero.

Where is it? A position needs a starting point

Ask someone where the bus stop is and they might say "200 metres down this street". That answer has three parts: a starting point (here), a direction (down this street) and a size (200 metres). Physics describes position the same way. We pick a reference point, call it zero, choose which direction counts as positive, and then give a number with a unit. On a straight road the position can be a single number such as x = +200 m. A car 50 m behind the starting line sits at x = −50 m.

The choice of reference point is up to you, and it changes the numbers but not the physics. If you sit in a moving train, the cup on your table is not moving relative to you, yet it is moving at the speed of the train relative to the ground. Both descriptions are correct. They simply use different frames of reference. OpenStax College Physics puts it this way: every motion is described relative to something, and you should always say what that something is.

Distance and displacement are different

Distance is the total length of the path you travelled. It only grows, and it has no direction. Displacement is the change in position: final position minus starting position, written Δx = x_f − x₀. It has a size and a direction, so it can be positive, negative or zero. Walk 5 m east and then 3 m west. The distance is 5 + 3 = 8 m. The displacement is +5 − 3 = +2 m, which means you end up 2 m east of where you started.

This is why a runner who finishes a lap where they started has a displacement of zero even though their legs did 400 m of work. A delivery driver who drives across town and back has a large distance on the meter but a displacement of zero at the end of the day. Distance tells you about the trip. Displacement tells you where you ended up compared with where you began.

Scalars and vectors

Quantities with only a size, such as distance, time, mass and temperature, are called scalars. Quantities with a size and a direction, such as displacement, velocity, acceleration and force, are called vectors. On a straight line we show the direction of a vector with a plus or minus sign. In two dimensions we draw an arrow: its length shows the size and its tip shows the direction. You will meet these arrows again in the motion lab, where the ball carries a velocity arrow and its two parts.

A quick check helps whenever you meet a new quantity: ask "does it make sense to say which way?" It makes sense to say a car moved 3 km north, so displacement is a vector. It does not make sense to say a lesson lasted 40 minutes north, so time is a scalar.

Maps and sat-navs use the same ideas in two dimensions. A walk of 300 m east and then 400 m north covers 700 m of distance, but the straight-line displacement from start to finish is only 500 m, pointing roughly north-east. That 500 m comes from Pythagoras: √(300² + 400²) = 500. A homing pigeon or a hiker with a compass cares about displacement; a pedometer counts distance. Whenever a question asks how far something is from where it started, it is asking about displacement.