The science
Stopping. Tyre friction caps the deceleration at amax = μ g. A brake command issued after a sensor delay τs and applied through a first-order actuator lag τ gives a(t) = −A(1 − e−t/τ), so the distance to rest is
s = v τ_s + v t* - A( t*^2/2 - τ t* + τ^2 (1 - e^(-t*/τ)) ), v = A( t* - τ(1 - e^(-t*/τ)) )
which tends to vτs + v²/(2μg) for a quick actuator. Speed enters squared - double the speed, four times the braking distance - and ice (μ ≈ 0.1) multiplies it by nine compared with dry asphalt.
Adaptive cruise (constant time gap). a = kp(gap − d0 − h v) + kv(vlead − v), limited to comfortable −3.5 .. +2 m/s² and to the cruise-speed controller. Following a lead at constant speed the gap settles at d0 + h vlead. Linearised with actuator lag the error obeys s³ + s²/τ + s(kv + h kp)/τ + kp/τ = 0; by the Routh test it is stable when kv + kp(h − τ) > 0 (a long brake lag needs a larger time gap).
Emergency braking (AEB). The system asks: what deceleration would stop me inside the gap plus the distance the lead needs to stop? areq = v² / (2(gap + vl²/(2μg))). When it passes 60 % of the friction limit it brakes at full grip (ideal ABS). The human model reacts after 1.2 s and brakes at 70 % of the limit, and only notices closing speed, not the lead's braking.
Simplifications: one-dimensional point-mass motion on a straight flat road, constant friction coefficient (typical textbook values, not a specific tyre), ideal ABS, no aerodynamic drag, no pitch/load transfer, perception modelled as a pure delay, radar noise ignored. The AEB trigger fraction, gains and reaction time are illustrative design choices, not a particular vendor's system. Do not use this model for real vehicle design.
Independent educational model; cars, cones and numbers are generic.