RPN: DM32 and DM42: Stopping Sight Distance (Metric)
The Stopping Sight Distance Formula – Derivation
The stopping sight distance (SSD) formula calculates the theoretical distance that a driver needs to see to react and stop to avoid colliding with a person or hazard safely.
The SSD is measured in meters (or in US units, feet). This blog entry will focus on the SI system (meters, kilograms, seconds).
The SSD is broken down into two parts:
SSD = d1 + d2
Part 1: d1: Reaction Distance
d1 = reaction distance = v * t
v = velocity of the vehicle
t = reaction time that the driver takes to hit their brakes. The ideal reaction time is 1 second (or less). However, if the driver is tired or is later in age, the reaction time will increase. Typically, the reaction time is assumed to be 2.5 seconds.
In calculating SSD, the velocity is entered in usually in km/hr. (kilometers per hour). We need to change this into m/s.
1 km / hr * 1,000 m / 1 km * 1 hr / 3,600 s = 1,000 / 3,600 m/s = 5 / 18 m/s
Note many publication rounds this conversion factor to 0.278.
Hence, the completed reaction distance portion is:
d1 = 5 / 18 * v * t
Part 2: d2: Stopping Distance
This part is more complicated and includes factors such as friction force (µ), weight of the car (mass/g, g = 9.80665 m/s^2), and grade of the road (grd%, which is the increase or decrease of the slope of the road).
A common formula for d2 is:
d2 = v^2 / (a * (µ + grd%))
Another way to determine d2 is to equate the kinetic energy of the car with the work required to stop the car:
KE = work
m * v^2 / 2 = (µ + grd%) * w * d2
where:
m = mass of the car, in kg
v = velocity of the car, in km/hr
µ = friction factor (unit-less)
d2 = distance in m
w = weight of the car in N
grd% = grade of the road, in decimal (i.e. 1% = 0.01) (unit-less)
g = 9.80665 m/s^2
Note that mass = weight / gravity acceleration; m = w / g:
w / g * v^2 / 2 = (µ + grd%) * w * d2
Solving for d2:
d2 = v^2 / (2 * g * µ) = v^2 / (2 * g) * 1 / (µ * grd%)
Note that d2 is in meters. But v is in km/hr. Once again, a conversion factor is required. I’m focusing on the portion v^2 / (2 * g). I’m going to break the problem down into two parts: numerator and denominator.
Numerator:
1 km^2 / hr^2 * 1^2 hr^2 / 3,600^2 s^2 * 1,000^2 m^2 / 1^2 km^2 = 25 / 324 m^2 / s^2
Denominator:
2 * g = 2 * 9.80665 m/s^2 = 19.6133 m/s^2
Numerator/Denominator:
(25 / 324 m^2/s^2) / 19.6133 m/s^2 ≈ 3.934090328 * 10^-3 m ≈ 1/254.188368 m
Publications and associations, such as the AASHTO (American Association of State Highway and Transportation Officials), will often round 254.188368 to 254.
The Completed Formula
SSD = d1 + d2 = d1 = 5/18 * v * t + v^2 / (254.188368 * (µ + grd%))
The value of µ usually takes the values between 0.3 and 0.4. For the program, I’m assuming that µ = 0.35 for wet road conditions and µ = 0.70 dry conditions
SSD in US Units
Using similar analysis, the SSD in US units is:
SSD = 22/15 * v * t + v^2 / (29.91388812 * (µ +grd%))
where: v = velocity in mi/hr (mph), t = reaction time in seconds, SSD in feet (ft)
Publications will round the constants to 1.47 and 30, respectively.
On obtaining the conversion factors, note that:
1 mi/hr = 22/15 ft/s
1 mi^2/hr^2 = 484/225 ft^2/s^2
2 * g = 2 * 9.80665 m/s^2 * 100/30.48 ft/s^2 ≈ 2 * 32.17404856 ft/s^2 ≈ 64.34809711 ft/s^2
(484/225) / (64.38409711) ≈ 0.033429288 ≈ 1/29.91388812
DM32/ HP 32II Program: Stopping Sight Distance (SI Units)
(not for the HP 32S because it uses messages)
D01 LBL D
D02 35 [store constants; wet conditions times 100]
D03 STO A
D04 70 [store constants; dry conditions times 100]
D05 STO B
D06 2.5 [store default reaction time]
D07 STO T
D08 INPUT T
D09 INPUT V
D10 INPUT G [enter grade as a percentage: 1% → 1]
D11 SF 10 [set message mode, SF, decimal point, 0]
D12 “1 WET 2 DRY”
D13 INPUT i [input indirect variable]
D14 CF 10 [turn off message mode, CF, decimal point, 0]
D15 RCL V
D16 x^2
D17 RCL (i)
D18 RCL+ G
D19 100
D20 ÷
D21 254.188368
D22 ×
D23 ÷
D24 5
D25 RCL× V
D26 RCL× T
D27 18
D28 ÷
D29 +
D30 STO D [store and view SSD]
D31 VIEW D
D32 RTN
HP 42S/DM42/Free 42 Program: Stopping Sight Distance (SI Units)
(This program is similar to the 32SII version.)
00 {121-Byte Program}
01 LBL “SSD”
02 35
03 STO 01
04 70
05 STO 02
06 2.5
07 STO 03
08 “REACT TIME?”
09 PROMPT
10 STO 03
11 RCL 04
12 “VELOCITY?”
13 PROMPT
14 STO 04
15 RCL 05
16 “GRADE?”
17 PROMPT
18 STO 05
19 RCL 00
20 “1. WET 2. DRY”
21 PROMPT
22 STO 00
23 RCL 04
24 X↑2
25 RCL IND 00
26 RCL+ 05
27 100
28 ÷
29 254.188368
30 ×
31 ÷
32 5
33 RCL× 03
34 RCL× 04
35 18
36 ÷
37 +
38 STO 06
39 “SSD=”
40 ARCL ST X
41 RTN
Variables:
R00 = choice variable
R01 = wet condition friction coefficient * 100
R02 = dry condition friction coefficient * 100
R03 = reaction distance (set to default as of 2.5 sec)
R04 = velocity (km/hr)
R05 = grade
R06 = SSD in meters
Examples
Velocity: 96.5606 km/hr (about 60 mi/hr), Time: 2.5 seconds, Grade: 0%
Dry Conditions (i = 2): SSD: 119.4578 m
Wet Conditions (i = 1, µ = 0.35), SSD: 171.8596 m
Velocity: 96.5606 km/hr, Time: 1.5 seconds, Dry Road
Grade: +1%: SSD: 91.8973 m
Grade: -1%: SSD: 93.3948 m
Sources
American Association of State Highway and Transportation Officials NCHRP Report 400. 1997. Last accessed April 27, 2025. https://onlinepubs.trb.org/onlinepubs/nchrp/nchrp_rpt_400.pdf
Chandra, Satish IITR “Stopping Sight Distance on a road. Definition, concept, and evaluation of SSD with examples.” YouTube Video. July 16, 2023. https://www.youtube.com/watch?v=HEzdJE7NQeU&t=973s Last accessed April 27, 2025.
Omni Calculator. “Stopping Distance Calculator” July 22, 2024. Last accessed April 26, 2025. https://www.omnicalculator.com/physics/stopping-distance
Eddie
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