Position, Reference Frames, and Displacement

Objective: Choose an origin and positive direction before describing motion.

A position is an object’s location relative to an origin in a specified coordinate system and reference frame. One-dimensional positions can be positive or negative according to the chosen direction; a negative coordinate does not mean a negative distance. Motion is described relative to a frame: a passenger can be stationary with respect to a moving train yet move relative to the ground. The choice of origin or positive direction changes numerical coordinates, but not the underlying physical event. Time must also be specified to compare positions. A position–time record is a model of the tracked object, often approximating it as a point when its size is not relevant.

Objective: Calculate path length and signed change in position.

Distance traveled is total path length and is a nonnegative scalar. Displacement is final position minus initial position, Δx = x_f − x_i, and in one dimension has a sign indicating direction. A trip from x = 2 m to x = 7 m and back to x = 4 m covers 5 + 3 = 8 m of distance but has displacement +2 m. For a closed trip, displacement is zero even when distance is positive. The magnitude of displacement cannot exceed distance traveled for a path between the same endpoints. Distinguish a coordinate from a change of coordinates and state the positive direction.

Objective: Distinguish magnitude-only quantities from quantities with direction.

A scalar is described by a magnitude and unit, such as distance, speed, elapsed time, or mass. A vector includes magnitude and direction, such as displacement and velocity. In one dimension, a sign can encode direction relative to a chosen positive axis; in two dimensions, components or an angle are needed for full specification. Two equal-magnitude displacements in opposite directions add to zero, whereas the corresponding traveled distances add. Vector addition follows geometry and chosen coordinates, not simply addition of positive magnitudes. The units of displacement and distance are both length units despite their different meanings.