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.

Reaction travel $\text{Ready}$

Attained braking travel $\text{Ready}$

Adhesion / force ceiling $\text{Ready}$

Power-to-force transition speed $\text{Ready}$

Selected active limit $\text{Ready}$

Half-step endpoint travel difference $\text{Ready}$

Endpoint outcome $\text{Ready}$

Selected time $\text{Ready}$

Selected travel $\text{Ready}$

Selected speed $\text{Ready}$

Selected retarding force $\text{Ready}$

Attained peak retarding force $\text{Ready}$

Selected brake / absorber work $\text{Ready}$

Endpoint kinetic energy $\text{Ready}$

Attained observation duration $\text{Ready}$

Attained total travel $\text{Ready}$

Model momentum balance residual $\text{Ready}$

Model energy balance residual $\text{Ready}$

Raw integrated net impulse $\text{Ready}$

Raw impulse-momentum residual $\text{Ready}$

Raw integrated net work $\text{Ready}$

Raw work-energy residual $\text{Ready}$

Acquired stations $\text{Ready}$

Stopping with impulse and work

What actually limits the stop?

Calibrated mass

m

1400.00\,kg

Initial forward speed

v_0

25.00\,\frac{\mathrm m}{\mathrm s}

Reaction delay

t_r

1.00\,s

Adhesion coefficient

\mu

0.75\,

Brake force limit

B_{\max}

7000.00\,N

Brake dissipation power limit

P_{\max}

100.00\,kW

Drag area

C_D A

0.65\,\mathrm{m^2}

Downhill angle; uphill is negative

\theta

0.00\,{}^\circ

Maximum observation time

T_{\mathrm{obs}}

60.00\,s

Integration time step

\Delta t

0.020\,\mathrm s

Selected time fraction

t/t_{\mathrm{end}}

0.40\,

Sensor sampling rate

f_s

20.00\,Hz

Sensor noise gain

\eta

0.00\,

The vehicle is a point mass on a straight road of constant slope, and a positive angle means downhill. During the reaction time it coasts under gravity and drag, with no cruise control or driving force.

Three limits cap the braking: tyre adhesion, actuator force and the power the brakes can dissipate. Constant adhesion stands for ideal grip control. Tyre slip, load transfer, wheel inertia, rolling resistance, brake fade and wind are left out.

The fourth-order Runge–Kutta integrator splits its step at the end of the reaction time and finds the moment of zero speed inside the final step. The run ends at the first moment of rest, before any rollback. If the time window ends while the car is still moving, the car has not stopped.

If the downhill pull exceeds the maximum brake force at low speed, the car can never come to rest in this model. Gravity can add kinetic energy while the brakes remove it. The difference in travel between full and half steps is a rough check, not a certified error bound.

An acquisition samples the whole time window actually reached, at the chosen sensor rate. The noise is the noise gain times one millimetre, one hundredth of a metre per second and one newton. Calibrated time and mass are exact, and raw values are never clipped.

Raw impulse and work are computed only from measured force, speed and time. Sampling can miss area under a sudden jump or a narrow peak in force. Whether the model's numbers agree is shown separately from whether the acquired record's numbers agree.

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

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 misconceptionIf the ideal model stops in time, a real cushion or road distance must be safe.

Better mental modelThis 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

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.
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.
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.
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.