Signed Displacement And Total Distance From Motion

Two Different Ways to Describe a Trip

A moving object can start and finish at positions that are close together after travelling a much longer route. To describe what happened, we need to distinguish two quantities: signed displacement, which records the net change in position, and total distance, which records how much ground the object travelled.

In one-dimensional motion, position is represented by a number on a line. The sign of a change in position tells us its direction: a positive change is motion in the positive direction, and a negative change is motion in the negative direction. Distance, in contrast, is never negative.

The distinction matters whenever an object changes direction. The starting and ending positions determine the displacement, but they do not necessarily tell us how far the object travelled in between.

Signed Displacement Uses Only the Endpoints

Let \(x(t)\) denote an object's position at time \(t\). Over the interval from \(t=a\) to \(t=b\), the signed displacement is the final position minus the initial position:

If the result is positive, the final position is in the positive direction from the initial position. If it is negative, the final position is in the negative direction. A zero result means the object ends where it started; it does not mean that the object stayed still.

Direction Changes Require Separate Intervals

To calculate total distance when direction changes, first split the motion into intervals on which the object moves in a consistent direction. A change from positive to negative velocity, or from negative to positive velocity, signals a reversal of direction. A momentary zero velocity is not by itself enough to establish a reversal: the object might stop briefly and then continue in the same direction.

On each direction-consistent interval, the object's position moves monotonically: it does not turn back along the line. The distance travelled on that interval is therefore the magnitude of the change between its endpoint positions. Add these nonnegative distances to get the total.

The partition points should include the times when the object reverses direction. They may also include times at which it pauses, provided each resulting interval remains direction-consistent. Adding a pause to the partition does not change the total distance.

Why the Two Calculations Differ

The difference between the two quantities follows from how they treat direction. Signed displacement retains the signs of the position changes. A movement in the negative direction subtracts from the net change. Total distance takes the magnitude of each change before adding, so movement in either direction contributes a nonnegative amount.

For a trip split at direction changes, write each successive position change as \(\Delta x_i=x(t_i)-x(t_{i-1})\). The endpoint displacement is the sum of the signed changes, because intermediate positions cancel:

On each direction-consistent interval, the distance travelled is \(|\Delta x_i|\). Thus total distance is the sum of those magnitudes, not the magnitude of the sum. Taking magnitudes before adding prevents travel in opposite directions from cancelling.

To see the inequality, use the triangle inequality for real numbers: the magnitude of a sum is no greater than the sum of the magnitudes. Applying it to the signed changes gives

This argument also explains the equality case. When all nonzero changes have the same sign, their magnitudes add to the magnitude of their sum, so distance equals the magnitude of displacement. Changes in opposite directions generally make the inequality strict.

Worked Example: Returning to the Starting Position

What Position Data Can and Cannot Tell You

Endpoint positions always determine signed displacement. They determine total distance only when the motion between them is known to be direction-consistent, or when enough information is available to identify all reversals and the positions at those times.

For example, if an object begins at \(0\) metres and ends at \(5\) metres, its displacement is \(5\) metres. If it moves steadily in the positive direction, its total distance is \(5\) metres. But the same endpoints are also consistent with a trip from \(0\) to \(8\) metres and then back to \(5\) metres; that trip has total distance \(8+3=11\) metres. Endpoint data alone do not distinguish these paths.

Likewise, a list of positions recorded at a few times does not necessarily reveal motion between observations. If an object passes through the same recorded positions while making an unobserved reversal between measurements, simply adding the magnitudes of the recorded position changes can underestimate its total distance. To use the sum correctly, the intervals must genuinely be direction-consistent.

Common Mistake: Adding Signed Changes for Distance

A frequent error is to calculate each change correctly but then add the signed changes and call the result total distance. In the round-trip example, that incorrect calculation would be

This is the signed displacement, not the distance. The negative sign records the direction of the return trip; it does not mean that the object travelled a negative length.

The correction is to take the magnitude of each interval's change before adding:

Another error is to assume that returning to the starting position means no motion occurred. Equal initial and final positions imply zero displacement, but the total distance is zero only if the object did not move at all.

A Reliable Calculation Process

For a one-dimensional motion problem, use the following sequence:

These checks do not replace the calculation, but they can reveal common errors. A negative total distance is impossible, and a reported distance smaller than the magnitude of displacement cannot describe the motion.