The 3D companion sim treats braking as an instant step: the moment system latency ends, deceleration jumps straight to its peak value. Real brake systems can't do that — hydraulic pressure build-up, ABS engagement and tire load transfer all take a real, measurable amount of time (documented in brake response-time testing such as ISO 21994) before full deceleration is reached. This 2D model replaces the instant step with a linear brake force build-up ramp of duration tb, which is genuinely different arithmetic, not a re-skin:
During buildup (0 ≤ t ≤ t_b): a(t) = a_max · (t / t_b)
v(t) = v0 − a_max·t² / (2·t_b)
x(t) = v0·t − a_max·t³ / (6·t_b)
If v(t_b) > 0 (car still moving after the ramp):
d_ramp = v0·t_b − a_max·t_b² / 6
v(t_b) = v0 − a_max·t_b / 2
d_const = v(t_b)² / (2·a_max) (constant a_max phase)
d_brake_ramped = d_ramp + d_const
If v(t_b) ≤ 0 (car stops mid-ramp):
t* = √(2·t_b·v0 / a_max)
d_brake_ramped = v0·t* − a_max·t*³ / (6·t_b)
Naive instant model (what the 3D sim uses): d_brake_naive = v0² / (2·a_max)
Build-up penalty = d_brake_ramped − d_brake_naive ≥ 0 always
Numerically verified against direct small-step (1 ms) numerical integration of the same a(t): closed-form and integrated stopping distance agree to within 0.1% across a range of speeds, build-up times and grip levels, and the closed form correctly reduces to the naive formula as tb → 0 (relative error ~2×10⁻⁵%). A full parameter sweep (20–140 km/h × 50–500 ms buildup × 2–9 m/s² grip) confirms the build-up penalty is never negative — physically, a slower-engaging brake can only ever cost distance, never save it.
- Brake build-up time — the new parameter here; longer build-up (soft or degraded hydraulics, cold ABS) always adds distance on top of both reaction and ideal braking distance, and the penalty grows faster than linearly as speed increases because the ramp phase covers more ground at higher v0.
- Why it matters — two vehicles with identical peak grip and identical system latency can still have meaningfully different real-world stopping distances if one has a slower-responding brake actuator; the naive v²/(2a) formula the 3D sim (and most textbooks) use silently assumes build-up time is zero.
- Speed and grip — same roles as the 3D model: reaction distance scales with v, ideal braking distance with v²/a; the build-up penalty stacks on top of both.