KAHIBARO
Table of Contents
Meaning of change in position
When an object moves in one dimension, we often want to describe how far it traveled and how its location changed. These are not the same idea. Distance and displacement both describe motion, but they answer different questions.
Distance tells us the total length of the path traveled. It is concerned only with how much ground was covered.
Displacement tells us the change in position from the starting point to the ending point. It includes direction, so in one dimension it can be positive, negative, or zero.
A person can travel a large distance and still have a small displacement, or even zero displacement, if they return to the starting point.
Distance
Distance is the total amount of motion along the path. In one dimensional motion, if an object moves forward and then backward, both parts add to the total distance.
Distance is a scalar quantity. That means it has magnitude only, with no direction.
For example, imagine a person walks 3 meters to the right, then 2 meters to the left. The total distance traveled is
$$
3 + 2 = 5 \text{ m}
$$
The path length always adds positively.
Distance is never negative.
Distance measures total path length, not change in position.
Displacement
Displacement compares the final position with the initial position. In one dimension, it is given by
$$
\Delta x = x_f - x_i
$$
where $x_i$ is the initial position and $x_f$ is the final position.
Displacement is a vector quantity. In one dimension, this means it carries a sign that shows direction relative to the chosen axis. A positive displacement means movement in the positive direction, and a negative displacement means movement in the negative direction.
Using the same example, if the person starts at $x = 0$, walks 3 meters to the right, and then 2 meters to the left, the final position is $x = 1$ m. So the displacement is
$$
\Delta x = 1 - 0 = 1 \text{ m}
$$
The displacement is 1 meter to the right.
Displacement is defined by final position minus initial position:
$$\Delta x = x_f - x_i$$
Displacement can be positive, negative, or zero.
Comparing the two
Distance and displacement often have different values. Distance depends on the entire trip. Displacement depends only on where the object started and where it ended.
| Quantity | Meaning | Type | Can be negative? |
|---|---|---|---|
| Distance | Total path traveled | Scalar | No |
| Displacement | Change in position | Vector | Yes |
A simple way to remember the difference is this. Distance asks, "How much did the object travel?" Displacement asks, "Where did the object end up compared with where it started?"
Examples in one dimension
Suppose a car starts at $x_i = 2$ m and ends at $x_f = 10$ m without turning around. Then the displacement is
$$
\Delta x = 10 - 2 = 8 \text{ m}
$$
If the motion was straight with no reversing, then the distance is also 8 m.
Now suppose the car starts at $x_i = 2$ m, goes to $x = 10$ m, then returns to $x_f = 6$ m. The distance is
$$
(10 - 2) + (10 - 6) = 8 + 4 = 12 \text{ m}
$$
But the displacement is only
$$
\Delta x = 6 - 2 = 4 \text{ m}
$$
So the car traveled 12 m, but its position changed by only 4 m.
Special case, returning to the start
If an object returns to its starting point, its displacement is zero, because the initial and final positions are the same.
For example, start at $x = 5$ m, move to $x = 9$ m, then come back to $x = 5$ m.
The distance is
$$
4 + 4 = 8 \text{ m}
$$
The displacement is
$$
\Delta x = 5 - 5 = 0
$$
This shows clearly that zero displacement does not mean zero distance.
An object can move a nonzero distance and still have zero displacement if it ends where it started.
Sign convention in one dimension
In one dimensional motion, we usually choose one direction as positive and the opposite direction as negative. For example, motion to the right may be positive, and motion to the left negative.
Because displacement depends on position, its sign depends on this coordinate choice. Distance does not depend on sign, because it is always positive.
Relationship between distance and displacement
The numerical value of displacement can never be greater than the distance traveled. In symbols,
$$
|\Delta x| \le \text{distance}
$$
This makes sense because the shortest possible path between start and finish in one dimension is the direct change in position. Any turning around increases distance but does not increase the magnitude of displacement.
For any motion,
$$
|\Delta x| \le \text{distance}
$$
Equality holds only when the object moves in one direction without reversing.
Why the distinction matters
This distinction is important because later motion quantities are built from these ideas. Average speed is based on distance, while average velocity is based on displacement. If distance and displacement are confused, later ideas about motion also become confusing.
For now, the key point is simple. Distance measures total travel. Displacement measures overall change in position. They are related, but they are not the same.