Stopping Sight Distance on a Grade (AASHTO)

Stopping sight distance is the distance a vehicle covers from the moment the driver sees a hazard to the moment it stops, and it is the sum of two parts. During the perception and reaction time $t_r$ the vehicle keeps moving at the speed $V$, so it travels $V\,t_r$ before the brakes do anything. Then it brakes at the deceleration $a$, covering $V^{2}/(2(a + gG))$. The grade $G$ is positive uphill: there gravity adds to the braking, so the braking distance is shorter, and downhill $G$ is negative, gravity works against the brakes and the distance is longer. Because the speed is squared in the braking term, that part grows faster than the reaction distance as the speed rises.

This is the AASHTO form, equation 3-2 of the Green Book as the Washington State Department of Transportation's design manual (M 22-01.23, exhibit 1260-3) prints it: $S = 1.47\,V\,(2.5) + V^{2}/(30[0.347826 \pm G/100])$ in feet and miles per hour. Here 2.5 seconds is the design perception and reaction time, and 0.347826 is the design deceleration of 11.2 ft/s² over 32.2. The metric Green Book form, $0.278\,V t_r + V^{2}/(254[a/9.81 \pm G])$ with $V$ in km/h, is the same relation with 3.4 m/s².

The page agrees with all 72 values of the manual's table of design stopping sight distances, for grades of 3, 6 and 9 percent up and down and speeds from 25 to 80 miles per hour, to within 0.5 percent; the table itself is rounded to whole feet. At 60 mph it gives 598 ft on a 3 percent downgrade, 686 ft on a 9 percent downgrade, 538 ft on a 3 percent upgrade and 495 ft on a 9 percent upgrade, so the grade changes the answer noticeably in both directions. On the level, at 100 km/h with 2.5 s and 3.4 m/s², it gives 183 m; the design table rounds such values up to a round number, for example 570 ft at 60 mph where the equation gives 566.

The page's own example takes the braking term alone. The Wikibooks text Fundamentals of Transportation, Example 1, has a vehicle at 66 km/h on a 3 percent downgrade with a friction coefficient of 0.3, so the deceleration is 2.94 m/s²; with the reaction time set to zero the page gives the 63.5 m the book finds. The relation is for a driver who brakes at the design deceleration: a wet road, a heavy truck or a driver who reacts late all lengthen the distance.