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Models and Proof

Deceleration Model | Driver Braking (g's) | Yellow Light Duration Formula
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.
Deceleration Model

model

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) ]

Solving for Yellow Duration (ty)
ty = tr - (Vo/2a) + di/Vo
Applies to flat level surfaces, see below for inclines.
Note, (a) is negative for deceleration (-a).
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).
a = af + g Gr
af = Acceleration (-af deceleration) on a flat surface
Gr = Road Grade Ratio, (+Gr) uphill, (-Gr) downhill
g = acceleration due to gravity constant (see below)


Accounting for inclines Acceleration becomes;
a = -vo / [ 2 (ty - tr - di/vo) ] + g Gr


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).
Braking g's = Bg = a / g
Braking g's

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

Yellow Light Time (ty)
ty = tr + { V / [ 2 ( a + g Gr ) ] } + di / V

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

Calculator -- Yellow Light Duration

Safe Decelerations (a)
10 - 11.2 ft/sec2
6.81 - 7.64 mph/sec
3.05 - 3.41 m/s2
10.97 - 12.29 kph/sec
0.31 - 0.35 g's

Some cities in Wisconsin reported to use 14 ft/sec2 (0.44 g's)

CopRadar.com/redlight/models
Red Light Camera Timing Issues

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