Signal Timing & Deceleration Requirement of MUTCD Table 4D-102

The Manual gives the following formula for Deceleration Distance.

1.                 D_d = ½ dt² or ½Vt or V²/2d.*

Where:

d     = deceleration rate of 10 ft/sec, (acceleration = -10 ft/sec²)**

V    = either a posted or a prima facie speed limit,

D_d   = deceleration distance.

t       = deceleration time.

*             The similarity of expressions such as dt to a derivative do not anywhere in this article refer to a derivative. They are used to retain similarity to the entries in the Manual.

The latter two expressions of this equation are achieved by substitutions of

t = V/d into ½ dt².

In the Manual the deceleration distance plus reaction time distance are added to generate a total stopping distance:

2.         D = V²/2d + Vt_R

Where t_R = reaction time, is assigned 1 second in the MUTCD and Vt_R is reaction time distance traveled.

The Manual then divides by V to achieve an equation for T, the yellow light change interval as follows:

3.         T = (V²/2d + Vt_R)/V = V/2d + t_R

For present purposes t_R, the reaction time, is set aside and only the braking time element T_b of the yellow light timing T is considered:

4.         T_b = V/2d

From the common expression V = at, velocity equals acceleration times time, in the form t = V/a we see that the deceleration yielded by dividing equation #2 above by V has doubled to 20 ft/sec²:

a = 2d = 2(-10 ft/sec²)***

The following example illustrates.

            At 30 mph, or 44 ft/sec, the time to stop (disregarding reaction distance) at a deceleration d of 10 ft/sec² is:

        V = dt,

        44 ft/sec = 10 ft/sec² (t),

        t = 4.4 seconds. (deceleration time only considered)

However the MUTCD final formula for yellow light duration, excluding reaction time, is:

T = V/2d = 44 ft/sec / 2(10ft/sec²) = 44/20 sec = 2.2 seconds.

In table 4D-102 (CA) the 1 second reaction time, is added producing the 3.2 second yellow interval for a 30 MPH speed limit.

From this result it can be concluded that 20 ft/sec² is has been judged an acceptable deceleration rate for use in the MUTCD and the 10 ft/sec² original nominal rate is merely a convenient waypoint for establishing the basis for the table of yellow light intervals.

***         Some web materials on this subject seem to have mistaken the equation in the manual for the general expression relating velocity to acceleration and time.