Displacement and distance are related, but they are not the same physical quantity. Displacement describes how far and in what direction an object ends up from its starting point. It depends only on the initial and final positions. Distance traveled is the total length of the route actually followed. It depends on the path.
This distinction matters in kinematics because two objects can have the same displacement but very different distances traveled. For example, an object can move along a long curved route and still finish near where it started. In that case, the distance traveled may be large while the displacement is small.
Displacement from coordinates
If the initial position is
\((x_0,y_0,z_0)\)
and the final position is
\((x,y,z)\),
then the displacement vector is
Displacement components.
\[
\begin{aligned}
\Delta x &= x - x_0 \\
\Delta y &= y - y_0 \\
\Delta z &= z - z_0
\end{aligned}
\]
Therefore,
\[
\begin{aligned}
\Delta \vec r &= \left(\Delta x,\ \Delta y,\ \Delta z\right)
\end{aligned}
\]
Its magnitude is the straight-line separation between the two positions:
\[
\begin{aligned}
|\Delta \vec r| &= \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}
\end{aligned}
\]
Distance traveled
Distance traveled is a scalar path length, usually written as
\(s\).
It satisfies
\[
\begin{aligned}
s \ge |\Delta \vec r|
\end{aligned}
\]
Equality holds only when the motion is straight-line motion from start to finish with no detours. That is why the calculator sets
\(s = |\Delta \vec r|\)
if no path length is entered. If you provide a larger path length, the calculator compares the actual route length with the straight-line displacement.
Average velocity and average speed
Let the elapsed time be
\(t\).
The average velocity vector is based on displacement:
\[
\begin{aligned}
\vec v_{\text{avg}} &= \frac{\Delta \vec r}{t}
\end{aligned}
\]
Its magnitude is
\[
\begin{aligned}
|\vec v_{\text{avg}}| &= \frac{|\Delta \vec r|}{t}
\end{aligned}
\]
The average speed is based on total path length:
\[
\begin{aligned}
\text{average speed} &= \frac{s}{t}
\end{aligned}
\]
Since
\(s \ge |\Delta \vec r|\),
average speed is always greater than or equal to the magnitude of average velocity.
Worked 3D example
Use the sample values
start \(=(0,0,0)\),
end \(=(3,4,12)\),
elapsed time \(t = 2\,\mathrm{s}\),
and path length \(s = 20\,\mathrm{m}\).
Step 1. Compute the displacement components.
\[
\begin{aligned}
\Delta x &= 3 - 0 = 3 \\
\Delta y &= 4 - 0 = 4 \\
\Delta z &= 12 - 0 = 12
\end{aligned}
\]
Step 2. Compute the displacement magnitude.
\[
\begin{aligned}
|\Delta \vec r| &= \sqrt{3^2 + 4^2 + 12^2} \\
&= \sqrt{9 + 16 + 144} \\
&= \sqrt{169} \\
&= 13\,\mathrm{m}
\end{aligned}
\]
Step 3. Compute the magnitude of average velocity.
\[
\begin{aligned}
|\vec v_{\text{avg}}| &= \frac{|\Delta \vec r|}{t} \\
&= \frac{13}{2} \\
&= 6.5\,\mathrm{m\,s^{-1}}
\end{aligned}
\]
Step 4. Compute the average speed.
\[
\begin{aligned}
\text{speed} &= \frac{s}{t} \\
&= \frac{20}{2} \\
&= 10\,\mathrm{m\,s^{-1}}
\end{aligned}
\]
This example shows the key idea: the route length is longer than the straight-line displacement, so the average speed is larger than the magnitude of the average velocity.
How to interpret the graph
The graph marks the start point, the end point, the straight displacement arrow, and an animated path. If you do not enter a separate path length, the route overlaps the displacement arrow. If you do enter a larger path length, the graph displays one representative bent path whose total length matches the value you entered. That visual route is not unique; many different paths could have the same total distance while sharing the same start and end points.
Summary
| Quantity |
Formula |
Meaning |
| Displacement components |
\(\Delta x = x - x_0,\ \Delta y = y - y_0,\ \Delta z = z - z_0\) |
Coordinate changes from start to finish |
| Displacement vector |
\(\Delta \vec r = (\Delta x,\Delta y,\Delta z)\) |
Vector from initial to final position |
| Displacement magnitude |
\(|\Delta \vec r| = \sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}\) |
Straight-line separation |
| Distance traveled |
\(s \ge |\Delta \vec r|\) |
Total route length |
| Average velocity vector |
\(\vec v_{\text{avg}} = \Delta \vec r/t\) |
Displacement per unit time |
| Magnitude of average velocity |
\(|\vec v_{\text{avg}}| = |\Delta \vec r|/t\) |
Size of the average velocity |
| Average speed |
\(s/t\) |
Distance per unit time |
