---
title: "Braking Distance & Power Limits Interactive Lab | Sandbox Physics"
description: "How do reaction time, tyre grip and brake power each decide how far a car goes before it stops? During the reaction time the car covers ground at full speed."
canonical_url: "https://sandboxphysics.com/labs/braking-distance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/braking-distance/"
last_modified: "2026-10-27"
---

M049 · Stopping dynamics

# Braking Distance & Power Limits

**How do reaction time, tyre grip and brake power each decide how far a car goes before it stops?** During the reaction time the car covers ground at full speed. After that, the braking force is capped by whichever limit is smallest: what the tyres can grip, what the brakes can push, or brake power divided by speed. The power cap bites hardest at high speed. On a steep enough descent, where gravity's pull down the slope is at least the braking ceiling, the car never stops.

Split a stop into reaction travel and braking travel, then compare the limits set by grip, brake force and brake power on a sloping road. Check whether the raw sensor data balance impulse and work. If the brakes are too weak for a downhill slope, the car does not stop.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The dynamics are idealized and the sensors are simulated. The Lab states its stopping conditions and the limits of its approximations explicitly.

Physics tutorial

## Reaction, traction and brake power

**Background** To stop something, you have to change its momentum and give its kinetic energy somewhere to go.

**Why it matters** The timing of the force, the distance the apparatus travels and the separately integrated sensor data should all tell the same story.

Start with the essentials

- **Focus question**: What actually limits the stop?
- **One-sentence intuition**: A power-limited brake is weakest at high speed; as the car slows, its force grows until it reaches the force ceiling.

### Core mathematical model

#### Three limits on the brake force

$$
\begin{aligned}B_0&=\min(B_{\max},\mu mg\cos\theta)\\B(v)&=\min(B_0,P_{\max}/v),\quad v>0\end{aligned}
$$

At zero speed the limit is the force ceiling. The power setting caps how fast the brakes can turn kinetic energy into heat.

#### Equation of motion on the slope

$$
\begin{aligned}\dot x&=v\\m\dot v&=mg\sin\theta-B(v)-\frac12\rho C_D A v^2\end{aligned}
$$

A positive angle means downhill. There is no braking during the reaction time, but drag and the slope still act.

#### Reaction and braking distance

$$
\begin{aligned}d_{\mathrm{total}}&=d_{\mathrm{reaction}}+d_{\mathrm{brake}}\\d_{\mathrm{reaction}}&=v_0t_r,\quad d_{\mathrm{brake}}=\frac{mv_0^2}{2B_0}\end{aligned}
$$

The second line is a benchmark for a level road with no drag and a fixed brake force. It does not apply as written when power limits the brake or the road slopes.

#### Work done by each force

$$
\frac m2[v(t)^2-v_0^2]=W_{\mathrm{grade}}-W_B-W_D
$$

The work done by gravity along the slope can be positive or negative. Until the car first comes to rest, the brakes and drag only absorb energy. On a steep descent the kinetic energy does not have to fall.

#### Exact solution for a power-limited stop

$$
\begin{aligned}v_c&=P_{\max}/B_0\\t_b&=\frac{m(v_0^2-v_c^2)}{2P_{\max}}+\frac{mv_c}{B_0}\\d_b&=\frac{m(v_0^3-v_c^3)}{3P_{\max}}+\frac{mv_c^2}{2B_0}\end{aligned}
$$

When the initial speed is above the transition speed, on a level road with no drag, a power-limited phase hands over to a constant-force finish. The raw sensor integration is checked separately.

### Common difficulties

#### An ideal model is not a safety rating

**Typical misconception** If the ideal model stops in time, a real cushion or road distance must be safe.

**Better mental model** This is a teaching model with prescribed force laws and simulated sensors. Real materials, tyres and differences between drivers have to be checked against measurements.

### Run the experiment

1. 01

   #### Separate the reaction time

   Start with Constant-force benchmark and change reaction delay.

   **What to observe:** The distance covered during the reaction time is separate from the distance covered while braking.

2. 02

   #### Find which limit is in charge

   Compare Dry-road reference, Low adhesion and Low brake power.

   **What to observe:** The force plot and the active-limit readout show where the power limit hands over to the force limit.

3. 03

   #### Try the slopes

   Try Cannot stop downhill and Coast to rest uphill.

   **What to observe:** On a downhill slope, gravity can keep the car moving even while the brakes take energy out. Uphill, the model ends when the car first comes to rest, before it can roll back.

4. 04

   #### Compare model and sensors

   Open Sensors and diagnostics to acquire a record; change sampling, noise and the integration step under Vehicle and numerical settings.

   **What to observe:** How accurate the solver is and how accurate the sensor integration is are two separate questions.

Physical conclusion

You can only say something stopped if there was enough stroke or braking to do it and the speed actually reached zero.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)