# Nakafa Learning Content

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

Learn stopping distance as the sum of reaction distance and braking distance in uniformly accelerated motion.

---

## Stopping Distance Is More Than Braking Distance

Stopping distance is the total distance a moving object needs until it fully stops. For a vehicle, this distance does not only happen while the brakes are acting.

Before braking begins, the driver still needs reaction time. During that time, the vehicle keeps moving forward with its initial speed.

Component: MathContainer
Children:

```math
d_{\text{stop}}=d_{\text{reaction}}+d_{\text{brake}}
```

```math
d_{\text{reaction}}=vt_r
```

So stopping distance has two stages. The first stage happens when the driver notices danger but the vehicle has not started braking. The second stage happens after the brakes act and the vehicle slows to a stop.

| Part | What Happens | What Determines It |
| --- | --- | --- |
| Reaction distance | The vehicle still moves with its initial speed | Speed and reaction time |
| Braking distance | The vehicle slows until it stops | Initial speed and deceleration magnitude |

Component: StoppingDistanceLab
Props:
- title: Car Stopping Distance
- description: Change the initial speed to compare green reaction distance with orange
braking distance.
- labels: {
chooseSpeed: "Choose initial speed",
speed: <>Initial speed</>,
reactionDistance: <>Reaction distance</>,
brakingDistance: <>Braking distance</>,
stoppingDistance: <>Stopping distance</>,
viewLabel: "Car stopping distance visual",
}

## Braking Distance Comes from Uniform Acceleration

While the brakes act, the vehicle slows down. If the deceleration is treated as constant and the final velocity is zero, a uniformly accelerated motion equation can be used to find the braking distance.

Component: MathContainer
Children:

```math
v_t^2=v_0^2+2a\Delta x
```

```math
0=v^2-2b\,d_{\text{brake}}
```

```math
d_{\text{brake}}=\frac{v^2}{2b}
```

The quantity $$b$$ is the magnitude of the deceleration, so it is written as a positive value.

Visible text: The quantity is the magnitude of the deceleration, so it is written as a positive value.

This formula uses a simple assumption: the path is straight, braking deceleration is treated as constant, and the vehicle does not skid. If the road condition changes, the value of $$b$$ can change too.

Visible text: This formula uses a simple assumption: the path is straight, braking deceleration is treated as constant, and the vehicle does not skid. If the road condition changes, the value of can change too.

## Speed Makes Stopping Distance Grow Quickly

Reaction distance is directly proportional to speed. If the speed doubles, the reaction distance also doubles for the same reaction time.

Braking distance is more sensitive because it contains $$v^2$$. As speed increases, the motion energy that must be removed by the brakes grows much more quickly.

Visible text: Braking distance is more sensitive because it contains . As speed increases, the motion energy that must be removed by the brakes grows much more quickly.

Suppose the reaction time is $$1\text{ s}$$ and the braking deceleration magnitude is $$5\text{ m/s}^2$$.

Visible text: Suppose the reaction time is and the braking deceleration magnitude is .

| Speed | Reaction Distance | Braking Distance | Stopping Distance |
| --- | --- | --- | --- |
| $$10\text{ m/s}$$ | $$10\text{ m}$$ | $$10\text{ m}$$ | $$20\text{ m}$$ |
| $$20\text{ m/s}$$ | $$20\text{ m}$$ | $$40\text{ m}$$ | $$60\text{ m}$$ |
| $$30\text{ m/s}$$ | $$30\text{ m}$$ | $$90\text{ m}$$ | $$120\text{ m}$$ |

Visible text: | Speed | Reaction Distance | Braking Distance | Stopping Distance |
| --- | --- | --- | --- |
| | | | |
| | | | |
| | | | |

When speed rises from $$10$$ to $$20\text{ m/s}$$, stopping distance rises from $$20$$ to $$60\text{ m}$$. It is not just doubled because the braking part follows the square of speed.

Visible text: When speed rises from to , stopping distance rises from to . It is not just doubled because the braking part follows the square of speed.

## Safe Space Before a Vehicle Stops

A vehicle moves at $$20\text{ m/s}$$. The driver's reaction time is $$1\text{ s}$$ and the magnitude of the braking deceleration is $$5\text{ m/s}^2$$.

Visible text: A vehicle moves at . The driver's reaction time is and the magnitude of the braking deceleration is .

Component: MathContainer
Children:

```math
d_{\text{reaction}}=20(1)=20\text{ m}
```

```math
d_{\text{brake}}=\frac{20^2}{2(5)}=40\text{ m}
```

```math
d_{\text{stop}}=20+40=60\text{ m}
```

So the vehicle needs $$60\text{ m}$$ to stop under those conditions.

Visible text: So the vehicle needs to stop under those conditions.

That $$60\text{ m}$$ is not only the distance after the brakes are pressed. The first $$20\text{ m}$$ happens before braking starts, and the next $$40\text{ m}$$ happens while the vehicle is braking.

Visible text: That is not only the distance after the brakes are pressed. The first happens before braking starts, and the next happens while the vehicle is braking.