Position, Path Length and Displacement

Position. For an object that moves along a straight line, we set up a number line: we fix an origin $O$ (the point $x = 0$), a positive direction (conventionally to the right), and a unit of length. The position of the object at any instant is then simply the coordinate $x$ of the point on this line where the object is located at that instant — a positive value of $x$ means the object is on the positive side of $O$, and a negative value means it is on the negative side.

Path length (distance travelled). As the object moves from one instant to a later instant, it traces out an actual route along the line. The total length of this route — measured strictly along the path actually taken, including any back-and-forth reversals — is called the path length, or simply the distance travelled. Path length is a scalar quantity: it is always positive (or zero) and has no associated direction. If a body moves from $x = 2$ m to $x = 9$ m and then comes back to $x = 7$ m, the path length covered is $(9-2) + (9-7) = 7 + 2 = 9$ m, even though it ends up only 5 m from where it started.

Displacement. The displacement of the object between two instants of time $t_1$ and $t_2$ is defined as the change in its position:

where $x_1$ and $x_2$ are the positions at $t_1$ and $t_2$ respectively. Displacement is a vector quantity — along a straight line it has a magnitude (how far, in a straight line, the final position is from the initial position) and a sign that indicates direction (positive for a net shift in the positive direction, negative for a net shift in the negative direction). In the example above, the displacement is $x_2 - x_1 = 7 - 2 = +5$ m, regardless of the fact that the object actually travelled 9 m of path length to get there.

Path length is never less than the magnitude of displacement. Because displacement measures only the net change in position along the straight line, while path length accounts for the entire route (including any reversal), we always have

with equality holding only when the motion is strictly one-directional over the interval considered (the object never reverses direction). This inequality is the geometric root of the important comparison between average speed and average velocity taken up in the next section. …