Distance & Displacement

One of the most fundamental tasks is describing how objects move. Two central concepts in this description are distance and displacement. Though often used interchangeably in everyday language, they have distinct scientific meanings.

Distance

Distance is a scalar quantity. It measures the total length of the path traveled by an object, irrespective of direction. For example, if a particle moves along a winding trajectory from one point to another, the distance corresponds to the actual length of that trajectory. Since distance is scalar, it is always positive and carries no directional information. It is measured in units of length, such as meters (m) in the SI system.

The shortest distance between two points in uncurved space is a straight line. But sometimes, boundary conditions do not allow that and the trajectory of an object bends. One example is the path of a car from Washington DC to New York City which can be seen in the following image. The calculated distance by GrasHopper is around $370\,\mathrm{km}$, whereas the shortest distance would be much smaller.

Displacement

Displacement, on the other hand, is a vector quantity. It represents the straight-line change in position from an initial point to a final point, including direction. Mathematically, displacement is defined as the difference between the final position vector $\vec{r}_2$ and the initial position vector $\vec{r}_1$.

\[
\Delta \vec{r} = \vec{r}_2 - \vec{r}_1
\]

This vector difference is illustrated by the red vector in the image below.

This definition makes clear that displacement not only has a magnitude (the shortest distance between the two points) but also a direction. Because of its vector nature, displacement can be positive, negative, or zero depending on the chosen coordinate system.

A simple example illustrates the distinction:

• If a student walks 100 meters east and then 100 meters west along the same path, the distance covered is 200 meters.
• The displacement, however, is zero, since the student’s final position coincides with the initial one.

In experimental setups, this distinction is crucial. Measuring distance tells us about the total path length an object has traversed, which is often important when considering energy expenditure or material wear. Measuring displacement is essential for analyzing motion in terms of velocity, momentum, and other vector-based physical quantities.

In practice, experimental physicists carefully select which of the two is more relevant depending on the problem. For instance, in particle tracking inside a detector, displacement provides information about momentum and scattering angles, while distance can be critical for calculating energy loss along the trajectory.

Exercises