# Nakafa Learning Content

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URL: https://nakafa.com/en/subjects/physics/kinematics/displacement-distance
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/physics/kinematics/displacement-distance/en.mdx

Learn the difference between distance as path length and displacement as the change from initial to final position.

---

## Two Ways to Read a Trip

When an object moves from a starting point to an ending point, two quantities can sound similar but mean different things: **distance** and **displacement**.

**Distance** is the total length of the path traveled. If the path turns, every part of the turn is included. **Displacement** is the change in position from start to finish, so it only depends on the starting point, ending point, and direction.

```math
\begin{aligned}
s&=\text{path length} \\
\Delta \vec{r}&=\vec{r}_{\text{final}}-\vec{r}_{\text{initial}}
\end{aligned}
```

Here, $$s$$ means the total path length, while $$\Delta \vec{r}$$ means the change in position from start to finish.

Visible text: Here, means the total path length, while means the change in position from start to finish.

In two-dimensional motion, displacement is drawn as the straight line from the starting point to the ending point.

> Distance adds the path actually traveled. Displacement reads the change from the start point to the end point.

Component: DisplacementDistanceLab
Props:
- title: Path Length and Displacement
- description: Choose a route to compare the traveled path with the straight
displacement.
- labels: {
chooseCase: "Choose route",
modeLabels: {
straight: <>Straight</>,
turn: <>Turn</>,
return: <>Back to Start</>,
},
factLabels: {
distance: <>Distance traveled</>,
displacement: <>Displacement magnitude</>,
vector: <>Displacement vector</>,
meaning: <>What changes</>,
},
meanings: {
straight: <>Distance and displacement magnitude are equal.</>,
turn: <>Distance is longer because the car follows the turn.</>,
return: <>The car travels a path but ends where it started.</>,
},
viewLabel: "Distance and displacement visual",
}

## Detours Change Distance

Compare an object that travels from the same start point to the same end point in two ways. A straight route gives a distance equal to the displacement magnitude. A detour gives a longer distance, but the displacement is still read as the line from start to finish.

The detour route has a greater distance because the object travels along a longer path. But the displacement stays the same as long as the starting and ending points stay the same.

This is the main question to keep separate: distance asks how much path was traveled, while displacement asks where the object ended up compared with where it started.

## Distance Has No Direction

Distance is always zero or positive. It answers the question, "how long is the path traveled?" So distance does not include direction.

If an object moves $$4 \text{ m}$$ east and then $$3 \text{ m}$$ south, the distance is:

Visible text: If an object moves east and then south, the distance is:

```math
s=4 \text{ m}+3 \text{ m}=7 \text{ m}
```

The symbol $$s$$ is often used for distance traveled.

Visible text: The symbol is often used for distance traveled.

## Displacement Has Direction

Displacement is a vector quantity. That means displacement has magnitude and direction. For the example $$4 \text{ m}$$ east and then $$3 \text{ m}$$ south, the displacement magnitude is the straight line from start to finish:

Visible text: Displacement is a vector quantity. That means displacement has magnitude and direction. For the example east and then south, the displacement magnitude is the straight line from start to finish:

```math
|\Delta \vec{r}|=\sqrt{4^2+3^2}=5 \text{ m}
```

The direction of displacement does not follow the turns of the path. It follows the line from the starting point to the ending point.

## Returning to the Starting Point

If an object travels around and returns to the starting point, the distance is not zero because a path was still traveled. But the displacement is zero because the initial and final positions are the same.

| Trip | Distance | Displacement |
| :--- | :------- | :----------- |
| From start to finish by a straight route | equal to the straight line | line from start to finish |
| From start to finish by a detour | longer | still the line from start to finish |
| Around a loop back to the start | not zero | $$0$$ |

Visible text: | Trip | Distance | Displacement |
| :--- | :------- | :----------- |
| From start to finish by a straight route | equal to the straight line | line from start to finish |
| From start to finish by a detour | longer | still the line from start to finish |
| Around a loop back to the start | not zero | |

So, distance tells the length of the actual path traveled, while displacement tells the net change in position.

## Winding Route with a Straight Result

A student walks $$8 \text{ m}$$ east, then $$6 \text{ m}$$ north. The distance traveled is:

Visible text: A student walks east, then north. The distance traveled is:

```math
s=8 \text{ m}+6 \text{ m}=14 \text{ m}
```

The displacement magnitude is:

```math
|\Delta \vec{r}|=\sqrt{8^2+6^2}=10 \text{ m}
```

The distance is $$14 \text{ m}$$, but the displacement magnitude is $$10 \text{ m}$$. The values are different because the path traveled is not the same as the straight line from start to finish.

Visible text: The distance is , but the displacement magnitude is . The values are different because the path traveled is not the same as the straight line from start to finish.