Deceleration Model
The braking (deceleration) model determines the point a vehicle must stop based on speed, reaction time, yellow duration, and distance between the stop line and camera trigger. The model also determines the braking (deceleration) for a vehicle whose driver decides to stop.
| ||
Proof | ||
GIVEN Vo = Speed tr = driver reaction time ty = yellow light duration di = distance between stop line and photo line |
-- |
SOLVE FOR dr = distance traveled during react time db = distance traveled during braking time tb = braking time -a = deceleration |
Four unknowns (dr, db, tb, a) and 4 equations. | |||
From Layout db = dy - dr - di db = Voty - Votr - di db = Vo(ty - tr) - di | -----> |
From Newton's Law of Motion db = Votb + 0.5 a tb2 db = Vo tb + 0.5 (-Vo / tb) tb2 db = Vo tb - 0.5 Vo tb db = 0.5 Vo tb tb = 2 db / Vo |
From Layout a = -Vo / tb <----- |
By Definition dr = Vo tr dy = Vo ty |
tb = 2 (Vo ty - Vo tr - di) / Vo
tb = 2 (ty - tr - di/Vo) a = - Vo/tb = -Vo / [ 2 (ty - tr - di/Vo) ] |
INCLINES
Inclines (because of gravity) effect braking by making it easier to brake uphill, and harder downhill. The difference is a function of gravity and the ratio of elevation change to horizontal distance (road grade ratio Gr). Gr is positive for uphill (+Gr), negative for downhill (-Gr), and zero for level (Gr = 0).
Driver Braking (g's)
Deceleration can be equated to the Gravitational Constant (g), acceleration due to gravity. By International definition 1 g is a change in speed (acceleration, sometimes mistakenly confused with force) of exactly 9.80665 meters per second every second (32.17405 ft/s2).
Bg = Braking g's v = Speed tr = Driver Reaction Time ty = Yellow Light Duration |
di = Distance Between Stop Line and Photo Line Gr = Road Grade Ratio, +Gr uphill, -Gr downhill g = Gravitational Constant = 9.80665 m/s2 = 32.17405 ft/s2 |
Yellow Light Duration Formula
ty | = | tr | + | V ---------------- |
+ | di ----- |
2 (a + g Gr) | V |
ty = Yellow Light Duration tr = driver Reaction Time V = initial Speed a = Deceleration |
di = Distance from Stop Line to Photo Line Gr = road Grade ratio, +Gr uphill, -Gr downhill g = Gravitational Constant |