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How to Safety Rate Red Light Cameras
Rate Stoplights with Driver Braking

Models and Proof

Mathematical Model

The braking (deceleration) model determines the point a vehicle has just enough time and distance to continue through the traffic signal light based on speed, reaction time, yellow duration, and distance between the stop line and clear point. 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 clear point 		-- 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

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Driver Braking (g's)

Deceleration (-a) can be equated (compared) 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).

Rate a Traffic Stoplight to Driver Braking (g's) by
  • measure yellow light duration,
  • measure di (Stop/Clear Line Distance),
  • determine speed limit,
  • determine road grade (incline),
  • select a reaction time considered safe.

    • gb = Braking g's = a / g
    gb = -V / [ 2 g (ty - tr - di/V) ] + Gr =
    -V
    		 + Gr
    2 g (ty - tr - di/V)

    gb = Braking g's
    V = Speed
    tr = Driver Reaction Time
    ty = Yellow Light Duration     		di = Distance Between Stop Line and Clear Point
    Gr = Road Grade Ratio, +Gr uphill, -Gr downhill
    g = Gravitational Constant = 9.80665 m/s2 = 32.17405 ft/s2

    English and Metric Units Conversion
    ty & tr in seconds, Gr = ratio: +Gr Uphill, -Gr Downhill
    ENGLISH UNITS (g = 32.17405 ft/s2)
    V in mph, di in feet
    gb = -v / [ (15/11) g (ty - tr - (15/22) di/v)] + Gr
    gb = -v / [ 43.8737 (ty - tr - 0.6818 di/v)] + Gr

    METRIC UNITS (g = 9.80665 m/s2)
    V in kph, di in meters
    gb = -v / [ 7.2 g (ty - tr - 3.6 di/v)] + Gr
    gb = -v / [ 70.6079 (ty - tr - 3.6 di/v)] + Gr

    Calculator -- Braking g's


    Examples of Braking versus Yellow Light Duration (di = 0)
    -- Graph for reaction time of 1.5 sec
    -- Graph for reaction time of 2.3 sec
    -- Table for reaction time of 2.5 sec

    Example of
    Stoplight Data & Safety Rating
    Location: MADISON AVE at 27th St.
    Granite City, IL
    RED LIGHT CAMERAs
    Speed Limit: 35 mph
    Road Grade: 0 %
    Stop-Clear (di): 10 feet
    Yellow Time: 4.0 seconds (Video & Data)
    Braking* = 0.53 g's / Unsafe**
    * Based on Average Driver Reaction Time tr = 2.3 seconds.
    ** Driver Maximum Braking -- see Braking Scale below.

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    Braking Scale

    Traffic engineers commonly use for safe braking decelerations that range from 10 - 11.2 ft/sec2 (0.31 - 0.35 g's). Maximum safe braking (maintain control with good to excellent tires on a dry surface) varies from 15 ft/sec2 (0.47 g's) for an average driver to 20 ft/sec2 (0.62 g's) for a moderately skilled driver. Vehicle radar (crash avoidance) uses automatic braking that ranges from 3 - 5 m/sec2 (0.3 - 0.5 g's). Production vehicle maximum braking varies with make/model/year from 0.72 - 0.94 g's. One manufacturer claims a deceleration of 9.8 m/sec2 (1.0 g's), reports of vehicles stopping at 1.3 g's cannot be confirmed.
    Braking Scale Rank
    Rank g's BRAKING
    1 0.00 - 0.30 light
    braking    		 S
    A
    F
    E
    2 0.30 - 0.35 MODERATE*
    3 0.35 - 0.45 Hard U
    N
    S
    A
    F
    E
    4 0.45 - 0.65 DRIVER MAXIMUM
    Average to
    Skilled Driver
    5 0.65 - 0.70 Dangerous
    6 0.70 - 0.95 VEHICLE MAXIMUM
    varies with
    make/model/year...
    7 > 0.95 EXTREME
    * Used by traffic engineers as safe braking.     		Gravity Constant and Acceleration
    g's m/s2 ft/s2 mph/sec kph/sec knots/sec
    0.10 0.98 3.2 2.2 3.5 1.9
    0.20 1.96 6.4 4.4 7.1 3.8
    0.30 2.94 9.7 6.6 10.6 5.7
    0.40 3.92 12.9 8.8 14.1 7.6
    0.50 4.90 16.1 11.0 17.7 9.5
    0.60 5.88 19.3 13.2 21.2 11.4
    0.70 6.86 22.5 15.4 24.7 13.3
    0.80 7.85 25.7 17.5 28.2 15.3
    0.90 8.83 29.0 19.7 31.8 17.2
    1.00 9.81 32.2 21.9 35.3 19.1

    Also see;
    -- Acceleration Conversions Table & Calculator.
    -- MEMBERS SECTION Appendix A -- Acceleration Parameters.

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    Yellow Light Duration Formula

    The yellow light duration (ty) formula is;

    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 Clear Line
    Gr = road Grade ratio, +Gr uphill, -Gr downhill
    g = Gravitational Constant

    Calculator -- Yellow Light Duration


    Typical 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)

    Some places have been reported to not account for differences between stop line and clear point (di = 0). Also see Factors/Distance from Stop Bar to Clear Line.

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    © Copyright 2005, 2010, 2011 Donald S. Sawicki, All Rights Reserved.