APPENDIX A

Appendix A explains how to derive an adjustment factor for non-preventable crashes in Chapter V Benefits. Chapter V uses the change in delta v to estimate benefits from correcting the under-inflated tire pressures.  Change in delta v at a given traveling distance is defined to be one half of the velocity difference between two scenarios: a vehicle with correct tire pressure and without.  The total traveling distance of a vehicle after braking under the correctly inflated tire pressures is defined as the correct stopping distance.  The incorrect stopping distance is the total traveling distance of a vehicle with under-inflated tire pressures.  Change in delta v increases with the traveling distance.  Figure A-1 depicts a simplified curve relationship between change in delta v and the traveling distance for illustration.  The curvature varies with initial speed, deceleration, and traveling distance.  For non-preventable crashes, the maximum change in delta v occurs at the correct stopping distance.  Therefore, applying the change in delta v at this level to the total applicable baseline population would overestimate the benefits from correcting tire pressures for non-preventable crashes.

Figure A-1 Generalized Relationship Between Change in Delta V and Traveling Distance

Ideally, the benefits would be estimated by applying the change in delta v at any given traveling distance to the corresponding baseline population.  However, the change in delta v varies with the initial traveling speeds, deceleration, and traveling distance.  There are too many initial traveling speeds and deceleration (or stopping distance) combinations to be exhaustively analyzed.  Less ideally, the benefits would be estimated by applying the expected changes in delta v to the applicable baseline population.  This approach, too, encounters the same obstacles as in the ideal approach.  In addition, the expected changes in delta v might be fractions of 1 mile/hour, e.g., 0.1, 0.01 mile/hour.  This would result in infinite ways to segment the measurement units for the delta v based injury probability curves.  Thus, this analysis only uses a weighted average initial speed, correct stopping distance, and incorrect stopping distance to estimate the expected change in delta v with respect to the traveling distance.  The expected change in delta v can be considered as the mean for continuous variables (i.e., traveling distance). The adjustment factor is the ratio of the expected change in delta v and change in delta v at the correct stopping distance.  The following two sections describe the process in detail. In the last section, the sensitivity analysis examines several scenarios to estimate the impact of different initial traveling speeds, decelerations, and stopping distances on the adjustment factors.  Note that all the equations and functions were derived assuming constant decelerations.

Expected Change in Delta V

The expected change in delta v is an integral of the product of two functions: the probability density function of a non-preventable crash occurrence and the change in delta v at any given traveling distance d.  The change in delta v function is one half of the velocity change function.  The expected change in delta v (EDV) is:

                     -------- (1)

Where, SD_c= the correct stopping distance

The function  is the change in delta v by definition.  Note that the measurement unit among variables has to be consistent for Equation 1 and the rest of the equations.  For example, if "feet-second" measurement is used, then the velocity, deceleration, and traveling distance have all been based on feet-second unit.

   Probability Density Function

Assuming that the non-preventable crashes occurred uniformly at any traveling distance between the initial braking (d=0) and the correct distance, the probability density function u(d) has the property that . By solving this equation, u(d) is a constant function:

              --------- (2)

where, SD_c = the correct stopping distance

Change in Delta V Function

Change in delta v function between two tire pressure conditions is half of the velocity change function at any traveling distance d.  The velocity change function is

          -------- (3)

Where, v_i = velocity with incorrect tire pressure

SD_c = the correct stopping distance

At any given traveling distance d, the velocity under a constant deceleration can be derived based on the following formula: , where v_o is the initial traveling speed and a the deceleration.  Let variables a_i and a_c represent the deceleration under incorrect and correct tire pressure, respectively.  Then,

                        -------- (4)

At the stopping distance, a braking vehicle has 0 velocity.  Thus, for incorrect tire pressure:

and

               --------(5)

where:

Similarly, the deceleration formula for braking vehicles with the correct tire pressure is

              -------- (6)

where:

By substituting the right side of Equations 5 and 6 for a_i and a_c into Equation 4, the velocity change function can be rewritten as a function of initial traveling distance.

-------- (7)

Change in delta v at any given traveling distance d would be

             ------- (8)

For passenger cars, the weighted average initial traveling speed, correct stopping distance, and incorrect stopping distances are:

V₀ = 45.078 mile/hour = 66.114 feet/second

SD_c = 85.273 feet

SD_i = 86.464 feet

At any given traveling distance, the change in delta v is calculated by substituting these numbers into Equation 8.  Figure A-2 shows the change in delta v by traveling distance.   At the correct stopping distance of 85.273 feet, for example, the change in delta v DV(85.273) is:  

Figure A-2. Change in Delta V by Traveling Distance

Passenger Cars

For light trucks and vans, the weighted average initial traveling speed, correct stopping distance, and incorrect stopping distances are:

V₀ = 45.078 mile/hour = 66.114 feet/second

SD_c = 90.726 feet

SD_i = 91.979 feet

Figure A-3 shows the change in delta v by traveling distance.   At the correct stopping distance of 90.726 feet, the change in delta v DV(90.726) is:

Figure A-3. Change in Delta V by Traveling Distance

Light Trucks/Vans

After calculating the change in delta at the correct stopping distance, the expected change in delta v must be derived to calculate the adjustment ratio.

Expected Change in Delta V

The expected change in delta v is an integral of the product of the probability density function of a non-preventable crash occurrence and the change in delta v at any given traveling distance d.  The crash probability density function (Equation 2) is a constant function as described in the previous section.  With known correct and incorrect stopping distances and under a constant deceleration condition, the change in delta v function is a function of the traveling distance (Equation 8).  Substituting these two equations back to Equation 1, the expected change in delta v function can be rewritten as:

Where, c₀ and c₁ are constants.

For passenger cars, the expected change in delta v is:

Adjustment Factors

The adjustment factor is the ratio of expected change in delta v and change in delta v at the correct stopping distance, i.e.

For passenger cars, under the following set of conditions:

For light trucks and vans, under the following set of conditions:

Sensitivity Study

The sensitivity study examines the variations of the adjustment factors under 12 different scenarios - combinations of three initial traveling speeds (35, 49, and 62 mph), two vehicle types (passenger cars, light trucks/vans), and two roadway conditions (dry, wet).  Table A-1 lists the criteria of these 12 scenarios and the associated stopping distances and case weights.  Readers can refer to Chapter V for detailed explanations on how the initial traveling speeds, stopping distances, and weights were derived for these 12 scenarios.  Table A-1 also lists the calculated change in delta v at the correct stopping distance, the expected change in delta v, and the adjustment factor for each scenario.  The adjustment factors range from 6 to 10 percent.  It’s not surprising that the adjustment factors are smaller for dry pavement roadways.  As expected, the overall weighted adjustment factor is about 7 percent which equals to the overall 7 percent.

Table A-1.  Adjustment Factors and Related Statistics

	Initial Traveling Speed (mile/hour)	Correct Stopping Distance (feet)	Incorrect Stopping Distance (feet)	Change in Delta V at the Correct Stopping Distance (mile/hour)	Expected Change in Delta V (mile/hour)	Case Weights	Adjustment Factor
Passenger Cars, Dry Pavement
1	35	46.430	46.840	1.639	0.093	0.2920	0.06
2	49	91.097	91.926	2.324	0.135	0.3404	0.06
3	62	142.364	143.671	2.956	0.172	0.1110	0.06
Passenger Cars, Wet Pavement
4	35	53.389	54.587	2.594	0.223	0.1245	0.09
5	49	114.159	117.045	3.845	0.348	0.0963	0.09
6	62	202.678	208.722	5.276	0.511	0.0359	0.10
Light Trucks/Vans, Dry Pavement
7	35	49.120	49.551	1.630	0.093	0.2920	0.06
8	49	96.391	97.260	2.315	0.133	0.3404	0.06
9	62	150.704	152.076	2.947	0.170	0.1110	0.06
Light Trucks/Vans, Wet Pavement
10	35	57.552	58.817	2.566	0.219	0.1245	0.09
11	49	123.065	126.110	3.806	0.341	0.0963	0.09
12	62	218.405	224.748	5.206	0.499	0.0359	0.10