M049 · Stopping dynamics

Braking Distance & Power Limits

Separate reaction travel from braking, compare adhesion, force and power limits on a graded road, and audit raw sensor balances. An insufficient downhill brake is allowed to fail to 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 $\text{Ready}$

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 closure residual $\text{Ready}$

Raw integrated net work $\text{Ready}$

Raw work closure residual $\text{Ready}$

Acquired stations $\text{Ready}$

IMPULSE / WORK / LIMITS

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\,

Point-mass vehicle moves along a straight constant-grade road. Positive angle is downhill. During reaction, it coasts under gravity and drag; no cruise control or driving force is added.

Braking is limited by adhesion, actuator force and dissipation power. Constant adhesion represents ideal grip control; tyre slip, load transfer, wheel inertia, rolling resistance, brake fade and wind are omitted.

Fourth-order Runge–Kutta splits the reaction boundary and locates zero speed inside the final step. The model ends at first rest, before rollback. A finite window ending in motion is not a stop.

A downhill force exceeding the low-speed brake ceiling makes rest unreachable in this model. Gravity may add kinetic energy while brakes dissipate it. Half-step travel difference is a diagnostic, not a certified error bound.

Acquire the full attained window at the selected sensor rate. Noise gain scales one millimetre, one hundredth metre per second and one newton. Calibrated time and mass are exact; raw values are never clipped.

Raw impulse and work use only measured forces, speed and time. Sampling a force jump or narrow peak can miss area; agreement of the model is separate from agreement of the acquired record.

Idealized dynamics and simulated sensors; explicit stopping and approximation boundaries.

Start with the essentials

Focus question: What actually limits the stop?
One-sentence intuition: Constant-power braking is weakest at high speed, then reaches its force ceiling.

Core mathematical model

Three simultaneous brake limits

\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 limiting value is the force ceiling. The power control caps dissipative brake power.

Road dynamics

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

Positive angle is downhill. Braking is zero during reaction; drag and grade remain active.

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 level-road, zero-drag, force-limited benchmark. It does not apply unchanged to power limitation or a grade.

Track all work channels

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

Grade work is signed; brake and drag absorption are nonnegative before first rest. Kinetic energy need not decrease on a steep descent.

Power-limited analytic benchmark

\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}

For initial speed above the transition, zero grade and zero drag, power-limited motion joins a constant-force finish. Raw integration is audited separately.

Common difficulties

Scope of the idealization

Typical misconceptionAn ideal stopping model certifies a real cushion or safe road distance.

Better mental modelThis is a teaching model with prescribed laws and simulated sensors; real materials, tyres and driver variation require measured validation.

Run the experiment

01
Separate the delay
Start with Constant-force benchmark and change reaction delay.
What to observe: Reaction travel is distinct from distance under active braking.
02
Find the active limit
Compare Dry-road reference, Low adhesion and Low brake power.
What to observe: The force plot and selected limit identify the power-to-force transition.
03
Challenge the slope
Try Cannot stop downhill and Coast to rest uphill.
What to observe: Positive downhill grade can sustain motion despite dissipative braking. The uphill model ends before rollback.
04
Compare model and sensors
Acquire raw sensors, change sampling and noise, then halve the integration step.
What to observe: Integrator accuracy and sensor integration accuracy are independent diagnostics.