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4 Vehicle mass and size

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Introduction
Since the early 1970s, research established that drivers of larger, heavier cars have lower risks in crashes than drivers of smaller, lighter cars. - The effects are large and have been examined in detail in many studies. Examining effects in increasing detail has added to knowledge on a number of broad safety questions. This is why we devote this complete chapter to how vehicle mass and size affect safety. Although mathematical details are given for some topics, it is not necessary to follow the mathematics to grasp the main ideas and their applicability to other safety matters.
Vehicle factors
The term vehicle factors refers to physical attributes of a vehicle that affect risks. The term is most often applied to factors that influence risk to occupants when crashes occur, but may also refer to factors that affect the risk of crashing, such as the presence of antilock brakes or the height of the vehicle's center of gravity. The overriding concept is a difference in outcomes related to vehicle attributes if all other factors, especially driver behavior, are the same. This chapter focuses mainly on how driver risk is affected by the mass and size of vehicles, given that a crash occurs.
It is difficult to determine how any individual factor influences traffic safety because nearly all factors occur in the presence of other factors that have important effects on outcomes. This is particularly so in the case of vehicle factors, because the crash experience of a particular set of vehicles is so intertwined with use and driver behavior factors. Any particular vehicle factor is likely to attract purchasers with driver characteristics different from those who purchase other vehicles. Sporty vehicles attract different drivers than more narrowly utilitarian vehicles. In real-world crashes, all other things are never equal, and indeed are often far from equal. Effects that might seem due to vehicle factors are often enormously confounded by driver characteristics and use patterns.
Deaths per million registered cars does not measure vehicle factors
Figure 4-1 shows that the number of driver deaths per million registered cars trends lower with increasing mass. Much clearer than the dependence on mass is the dependence on the number of doors. There is no reason why adding doors to a car should substantially affect occupant protection. This effect is due not to the addition of the doors, but because the life-style and risk-taking characteristics of drivers who choose two-door cars differ in so many ways from those choosing four-door cars. These effects are far larger than explainable in terms of differences in the distributions of age and gender associated with drivers of the different car types.

Figure 4-1. Car-driver deaths in all crashes per million registered cars for 1994-97 models during 1995-98. Insurance Institute for Highway Safety (IIHS) data.

Examining only rollover crashes (Fig. 4-2) reveals even larger differences dependent on number of doors. Rollover risk depends particularly steeply on driver behavior, particularly speed choice.
Figure 4-3 shows data for single-car crashes. The effect of mass is less systematic, but there is still a general average increase in risk as mass decreases. Nominally, Fig. 4-3 indicates that 2,500-2,999 pound cars have higher risks than cars that are either lighter or heavier. Yet there are clear theoretical reasons, backed up by much empirical evidence, that vehicle mass affects the risk when a crash occurs in a systematic continuous manner. The non-systematic features of the relations provide additional evidence of the large role of non-vehicle factors.
Figure 4-4 shows that deaths in smaller cars do not result exclusively from crashes with heavier vehicles. Indeed, over a third of the deaths in cars of all the mass categories are in single-car crashes.

Figure 4-4. The percent of all car-driver fatalities that result from single-car crashes. IIHS data.6

An examination of fatal crash involvements for the same travel distance found corresponding effects. The rate for two-door cars was 44% higher than the rate for four-door cars, and the rate for two-door sports utility vehicles was 50% higher than the rate for four-door sports utility vehicles. (p 179)
Rates are for drivers
All the rates in Figs 4-1 to 4-4 are for drivers. This is the appropriate measure because occupancy systematically increases with vehicle size. So, even if all vehicles had identical occupant protection and crash experience, larger vehicles would have more fatalities per vehicle because more occupants would, on average, be at risk in each crash (p. 47). While risks are different in different car seats (Fig. 3-15, p. 54), the risks closely scale. So it seems likely that, although rear-seat passengers have lower absolute fatality risks than front seat occupants, the relative risks to rear seat occupants would follow patterns similar to those in Figs 4-1 through 4-4.
Mass or weight?
An object's weight is the force of gravity upon it, while its mass is the amount of substance it contains, as indicated, for example, by how much force is required to accelerate it. In most circumstances there is little practical difference, even though, conceptually, weight and mass are distinct. The mass of a body is typically determined by weighing it (determining the force necessary to prevent it from falling). When moving at constant speed a vehicle's weight generates the rolling resistance that primarily determines fuel use. In outer space the car would need no fuel to continue to move at constant speed, while on the moon propulsion energy to maintain a constant speed on a flat hard road would be substantially less than on earth. When a vehicle accelerates, fuel is used primarily to overcome inertia, which is proportional to mass. When in motion, the vehicle has kinetic energy determined by its mass and speed. When its speed is reduced due to breaking or crashing, this kinetic energy is converted into other energy forms, mainly heat. So, when vehicles crash, mass is the relevant characteristic. If a heavy and a light vehicle collided in space, each vehicle would undergo a speed change governed by Newton's laws of motion. The speed changes would not be all that different from those on a road because the earth's gravity is not a major factor.
There are over 20 definitions of vehicle mass. The one coded in FARS data is derived from the Vehicle Identification Number (VIN) and is generally the curb mass,7(p 17) defined as the mass of the vehicle with standard equipment and a full complement of fuel and other fluids, but with no occupants or cargo. This mass is determined by the design of the vehicle. We do not know how much cargo or fuel (filling the fuel tank typically adds about 50 kg) was on board a vehicle when it crashes. We know the numbers, ages, and genders of occupants, but we do not know their masses. The mass of four large occupants can exceed the mass of four small occupants by over 250 kg. Such uncertainties are unlikely to generate important systematic biases in most analyses, but will add substantial random noise.
In most of what follows, vehicle size is characterized by curb mass. When any relationship is plotted versus curb mass, this does not mean that mass is the causal factor. It might be size, or some combination of size and mass. In a typical set of vehicles, heavier vehicles are larger; smaller vehicles are lighter. (We use heavier/lighter to denote larger/smaller mass).
Two-vehicle crashes
Of the 25,840 drivers killed in US traffic in 2001, 43% died in two-vehicle crashes compared to 49% in single-vehicle crashes (Table 3-3, p. 48). Two-vehicle fatal crashes tend to be studied more than single-vehicle crashes because more important information is available. For single-car crashes, little information is generally available on damage suffered by struck objects. For a two-vehicle crash, the injuries sustained in one vehicle provide information relating to the crash forces on the other.
Definitions for two-vehicle crashes
From a formal perspective, each of the vehicles involved in a two-vehicle crash can be considered to have a symmetrical role - they crash into each other. However, for expository clarity it is convenient to make an arbitrary distinction between them, using such terminology as:
vehicle1 = first, striking, bullet, subject, driven, or your vehicle
vehicle2 = second, struck, target, partner, or other vehicle.
Vehicle masses are designated by m1 and m2. It is convenient for the heavier of the two vehicles to be vehicle2, so we can define a mass ratio, m, for every crash between two vehicles of known mass as
4-1
Choosing vehicle2 to be the heavier insures that m is greater than one.
Consider a set of crashes with the same value of m, or with values of m confined to a narrow range. Assume that the total number of drivers killed in the lighter vehicles is N1 and that N2 drivers are killed in the heavier vehicles. A driver fatality ratio, R, can be defined as
4-2
The interpretation of R is remarkably assumption free - a simple count of driver fatalities in two clearly defined sets of vehicles. It is a measure of relative fatality risk in pairs of crashing vehicles, essentially regardless of driver behavior or vehicle use patterns. Higher risk driving by, say, drivers of heavier vehicles will increase the number of driver fatalities in heavier vehicles, but also in lighter vehicles into which they crash by a similar proportion. Higher risk driving affects the total number of fatalities, which affects the precision with which R can be determined, but not its expected value. R is affected by factors that affect survivability in crashes, such as systematically different belt-wearing rates or systematically different driver ages in vehicles of different mass.
Effect of mass in two-car crashes
The above definitions and equations apply to crashes between vehicles of any type that differ in mass. We now focus on one class of vehicle, namely cars (body type 1-10 in FARS). This is because curb masses are coded in FARS for cars, but not for other types of vehicles. Cars constitute less than half of the vehicles on US roads. About one fifth of two-vehicle crashes involve two cars. There were 3,288 fatalities in two-car crashes in 2001, 7.8% of all fatalities. Even though two-car crashes are not responsible for a major portion of fatalities, they are studied intensively because they lead to findings that increase understanding of general effects that apply to any type of vehicle involved in any type of crash.
Empirical findings
Many analyses using FARS data - have found that R and m for cars are related according to
4-3
The example in Fig. 4-5 is for crashes between pairs of cars with unbelted drivers crashing into each other head-on (principal impact points 11, 12 or 1 o'clock - Fig. 3-16, p. 55). Placing any restriction on both vehicles involved in two-vehicle crashes greatly reduces sample sizes. If, say, half of all crashed vehicles suffer frontal damage, filtering out all vehicles not sustaining frontal damage reduces sample sizes by 75%. The relationship in Fig. 4-5, based on 15,356 unbelted drivers killed in 13,162 crashes, gives l = 3.58 ± 0.05.

Figure 4-5. Fatality ratio, R, versus mass ratio, µ, for frontal crashes (both cars with principal impact point at 11, 12 or 1 o'clock).12 The relationship R = m l is the first law of two-car crashes (with l = 3.58 for the data shown). FARS 1975-1998.

Some examples from the 30 values of l reported in one study9 are included in Table 4-1. Requiring that each driver be of the same gender and similar age does not appreciably affect l, justifying including drivers of both genders and all ages in the other analyses.

Table 4-1. Illustrative values of the parameter l in Eqn 4-3.

Explanation based on Newtonian mechanics
Simple Newtonian mechanics of two objects crashing into each other can offer insight into Eqn 4-3. Consider two cars, car1 and car2 with masses m1 and m2 traveling at speeds v1 and v2 towards each other with their centers of gravity moving along the same straight line (Fig. 4-6). Assume that after they collide they remain locked together (this is equivalent to the collision being non-elastic) in one clump of mass M = m1 + m2 traveling at speed V along the same straight line. Applying the law of conservation of linear momentum gives

showing that the ratio of the delta-v values is simply the inverse of the mass ratio. Equations 4-4 through 4-7 apply to any vehicles, or for that matter, to any objects crashing into each other.
If the masses and initial speeds of both cars were identical, both would stop at their point of contact immediately after the crash (V = 0). Indeed, an observer might find it difficult to distinguish between one car crashing into an unbreakable mirror in front of a barrier (vertical, unmovable, etc.) and two identical cars crashing into each other. (In the mirror case, the "identical" vehicles would have steering wheels on opposite sides).
Relationship between delta-v and fatality risk. Figure 4-7 shows the fraction,
P, of unbelted drivers involved in all types of crashes coded in the National Accident Sampling System who were killed versus an estimate of the vehicle's Dv. The data for Dv < 114 km/h fit well the function
4-8
This simple rule of thumb provides an effective fit to the data but has the undesirable formal property of an unrealistic discontinuity at Dv = 114 km/h. Consider a two-vehicle crash in which the vehicles experience delta-v values of Dv1 and Dv2 (both values less than 114 km/h). The ratio of the risks of death, R, to the drivers in the two vehicles is immediately computed from Eqn 4-8 as
4-9
So, combining the relationship between fatality risk and Dv in Eqn 4-8 with fundamental Newtonian mechanics reproduces the firmly established functional form of Eqn 4-3. The closeness of the power 3.54 to the values of l in Table 4-1 (ranging from 3.45 to 3.80) is fortuitous as there is no reason to expect a calculation that ignores so many details to agree this closely with the empirical data.

Figure 4-7. Probability of death versus delta-v for unbelted drivers. Data from Ref. 13, fit based on rule of thumb in Ref. 14.

First law of two-car crashes
Explaining a major portion of empirical relationships in terms of Newtonian mechanics unambiguously identifies mass, as such, as the major causal factor in the difference in driver risk when vehicles of dissimilar mass crash into each other. The robustness of the relationship Eqn 4-3 together with its explanation from basic physical principles and a relationship between fatality risk and delta-v suggests that R = m l is a law, the first of two laws of two-car crashes.
The relationship in Fig. 4-5 indicates that if two cars differ in mass by a factor of two, the driver in the lighter car is 12 times as likely to be killed as the driver in the heavier car (23.58 = 12.0). Only about 1% of US two-car crashes involve a mass disparity as great as a factor of two. Half of two-car crashes in the US involve cars with masses differing by more than 20%. For a 20% mass disparity, the driver in the lighter car is almost twice as likely to be killed as the driver in the heavier car (1.203.58 = 1.92).
The above risk comparisons are essentially unaltered if the relationship derived by considering how fatality risk depends on delta-v, Eqn 4-9, is used instead of the empirical relationship between fatality risk and mass ratio. The Eqn 4-9 relationship can be used to infer results for cases for which direct empirical information is unavailable.
Application to crashes between cars and large trucks. It might seem intuitively reasonable to assume that if a car and a large truck crash head on, the mass of the car is so much less than the mass of the truck that the car's mass would have little influence on the car-driver's risk. However, risk increases so steeply with delta-v that this is not so. 
Consider a large 1,800 kg car traveling at 50 km/h crashing head-on into a 12,000 kg truck traveling at 50 km/h in the opposite direction. The equations in Fig. 4-6 show that after the (assumed inelastic) crash, the clump comprising both vehicles travels at 37.0 km/h in the direction in which the truck was traveling. The unaided eye would likely perceive the truck continuing at its prior speed undiminished by the impact. However, the truck does have a delta-v of 13.0 km/h, which poses little risk to its driver. The large car has a delta-v of 87.0 km/h. Now repeat this scenario with a small 900 kg car replacing the 1,800 kg car. The lighter car has a delta-v of 93.0 km/h, which is 7% larger than the delta-v of the heavier car. Since fatality risk depends so steeply on delta-v this translates into a substantial 27% higher risk in the lighter car. Note that this calculation considers only how the mass of the car affects its delta-v. Even if the truck were of infinite mass, so that both cars had identical delta-v values of 100 km/h, risk would be lower in the heavier car because it would also be larger. Empirical evidence does indeed indicate that in car-truck crashes, risk to car drivers increases more steeply than due to delta-v effects alone.7(p 103),
Application to crashes between cars and pedestrians. Even when cars strike pedestrians, the mass of the car influences the pedestrian's speed change. A 75 kg stationery pedestrian struck by an 1,800 kg car traveling at 50 km/h will experience a delta-v of 48.0 km/h. If the striking car is 900 kg, the delta-v becomes 46.2 km/h. Thus the pedestrian struck by the heavier car has a delta-v that is 4% larger. While Eqn 4-8 was derived from vehicle crashes, it seems plausible that it would give an order of magnitude estimate for pedestrian impacts also, thus implying that the pedestrian struck by the heavier car is, solely from considerations of Newtonian mechanics, about 15% more likely to die.
Effect of other crash and driver characteristics
The relative risk to each of the drivers involved in a right-side impact crash is plotted in Fig. 4-8. The car struck on the side has damage at principal impact points 2, 3 or 4 o'clock, the other has frontal damage at impact points 11, 12 or 1 o'clock. The fitted line,
4-10
is Eqn 4-3 with the parameter a added to reflect how the risk depends on an attribute in addition to mass ratio. When m =1, the parameter a measures how that attribute influences risk. The data in Fig. 4-8 imply that the driver in the right-side-impacted car is 4.53 times as likely to be killed as the driver in the front-impacted car when the masses of each are the same. For the risk to be equal in each car, m = (1/4.53)(1/3.47) = 0.647. Thus, if the car struck on the side is 55% heavier than the other car, both drivers are at equal risk.

Figure 4-8. The relative risks to the drivers involved in right-side impact two-car crashes. The right-side-impacted car has principal impact damage at clock points 2, 3 or 4; the other car at 11, 12 or 1. FARS 1975-1989.9

This same approach was also applied to determine how driver characteristics affect driver fatality risk, thus providing approximate indications of a number of effects that will be determined more precisely by other methods in later chapters. The results are summarized in Table 4-2.
The first two rows show that a driver in a left-impacted car is 10.08 times as likely to be killed as the driver in the car with frontal damage, compared to the 4.53 times ratio for the impact on the right side. If we make the plausible assumption that risk in the frontally-impacted car does not depend on whether it strikes the left or right side of the other car, these results imply the risk to the driver in a side-impacted car is 10.08/4.53 = 2.2 times as great when the impact is on the left compared to on the right. A study based on simple counts of fatalities finds the side-impacted driver to be 3.5 times as likely to die as the front-impacted driver in a right-side impact and 6.6 times as likely in a left-side impact for a 1.9 risk ratio. These values, together with the 2.6 and 2.7 ratios on page 55, support the interpretation that risk increases steeply the closer the occupant is to the point of contact.
The other values in Table 4-2 show that not wearing a belt, consuming alcohol, being female, or being older are all risk-increasing factors. The simple comparison for belt wearing overestimates the risk-reducing effectiveness of belts because of biases discussed in Chapter 11. Otherwise the effects corroborate those determined with higher precision in later chapters. All the results refer to the risk of death given that the crash occurs - the drivers compared were each involved in the same two-car crash.
Other vehicles
Table 4-3 shows relative risks when vehicles of different types crash into each other, with all types of crashes included. Quantitative mass estimates are available in FARS only for cars. When light cars and large trucks crash into each other, the driver in the light car is 44 times as likely to die as the truck driver. When heavy cars and large trucks crash into each other, the driver in the heavy car is 22 times as likely to die as the truck driver. If one assumes that the car-size has little influence on the truck driver's risk, this implies that the driver of the light car is about twice as likely to die as a driver of the heavy car, in agreement with other findings.16 When small cars and mopeds crash into each other, the moped driver is 139 times as likely to die as the car driver.

Table 4-3. Risk to driver in vehicle1 relative to the risk in vehicle2 when these two vehicles crash into each other. Based on Ref. 9 using FARS 1975-1989.

Interpreting risk ratios
The comparisons above are based on risk ratios - the risk to one driver divided by the risk to the other. While large risk ratios have been recognized for many decades, it is only more recently that the term vehicle aggressivity has been used. This term has been most commonly applied to crashes between cars and light trucks, especially sport utility vehicles (SUVs).17, For frontal crashes a ratio of 5 car-driver fatalities for each SUV-driver fatality is reported.17 The major portion of this difference arises because of a difference in average mass between the vehicles. When the mass factor is controlled, the car driver is about twice as likely to die as the SUV driver.
The following hypothetical example illustrates that larger risk ratios do not necessarily indicate lower safety. Suppose we start with two identical original vehicles. If they crash head-on into each other, each driver has identical risk, say equal to 1 in arbitrary units. Now suppose that one vehicle is replaced by a new vehicle that reduces risk to its occupants by 15%, but also reduces risk to occupants of any vehicle into which it crashes by 5%. The redesigned vehicle thus reduces risk to all occupants in any two-vehicle crash in which it is involved.
If new and old vehicles crash into each other, the risk ratio is 0.95/0.85 = 1.12, compared to a former value of 1.0. The driver of the old vehicle is now 12% more likely to die than the driver of the new vehicle, whereas formerly they had equal risks. Although the risk ratio to the driver of the older vehicle increased by 12%, it clearly does not imply the new vehicle is "more aggressive."
Available data could not uncover the properties hypothesized for this new vehicle. All that would be observed is that drivers of vehicles into which it crashed were at higher relative risks than before the design change. The literature is replete with inappropriate interpretations of risk ratios as meaning more than changes in relative risk, which in this case, would suggest that the new vehicle is reducing net safety when it is in fact increasing it.
Separating causal roles of mass and size
While robust relationships have been shown between various factors and vehicle mass, this does not mean that mass is the causal factor. Vehicle size also affects safety, and heavier vehicles tend to be larger. Size and mass both affect safety. One wants to separate the causal roles, especially as this could suggest vehicle design changes to improve safety.
The relationship between car size and car mass
Imagine a hypothetical world in which all cars are made from material of the same density. If cars were of identical shape, differing only by a scale factor, then mass would be proportional to any linear dimension to the power three. However, regardless of their size, cars must be of sufficient height to accommodate seated humans. If all cars had the same height, so that only length and breadth varied, then mass would be proportional to length (or breadth) to the power two. Real cars are likely to be intermediate between these two hypothetical cases, suggesting a relationship between mass and a linear dimension of the form
4-11
where m is the mass of the car and w is a linear dimension (other than height) and a and b are constants, with b expected to be between 2 and 3.
Figure 4-9 shows mass versus wheelbase (the distance between the front and rear wheels) for each of the 4,081 unique pairs of wheelbase and mass combinations for cars of all model years coded in 2001 FARS. Cars associated with more than one mass (because they are sold with choices of different engines, etc.) contribute more than one data point. However, the number of unique wheelbase-mass pairs reflects mainly the enormous variety of cars. Because mass is available only for cars of model year 1966 and later, 115 earlier models going back to model year 1930 are not included in Fig. 4-9. The fit to
Eqn 4-11 gives
4-12
thus validating our intuitive understanding that b should be between 2 and 3.

Figure 4-9. Relationship between car size (as measured by wheelbase) and car mass, based on 4,081 unique wheelbase-mass pairs coded in FARS 2001.

The value of b was investigated as a function of model year, with little indication of any obvious dependency. A lower value, b = 1.9 ± 0.2, is reported for the relationship between total car length and mass for 12 European car models. A value of b =2 is consistent with constant density and car height not increasing with increasing length. The larger value in Fig. 4-5 is consistent with height or density increasing with car mass.
Additional data on the close relationship between measures of size and
weight are indicated by correlation coefficients between curb weight and wheelbase, curb weight and trackwidth (the distance between the left and right wheels), and wheelbase and trackwidth of 0.93, 0.92 and 0.91. Any observed empirical relationship between any safety measure and one of these quantities is going to provide a similarly good relationship with any of the others. In multivariate analyses, simply as a result of Eqn 4-12, coefficients associated with length will be about 2.45 times as large as those associated with mass. Early suggestions that size is a more important causal factor than mass might have arisen simply because of larger regression coefficients being found for size than for mass.3
Second law of two-car crashes - crashes between cars of same mass
When cars of the same mass crash into each other, Eqn 4-3 provides no useful information. However, Fig. 4-10 shows that five sets of data , and a calculated relationship support that the relative driver risk, RMM, when two cars of the same mass, M, crash into each other is given by
4-13
where k is a constant.12 Although the relationship is in terms of mass, it is size that is the causal factor. Mass is irrelevant to the Newtonian mechanics of two cars of the same mass crashing into each other. Further evidence that, when cars of similar mass crash into each other, driver fatality risk is proportional to the common mass is provided by regression relationships for 1991-1999 model-year cars.7(p 103) Reducing the masses of cars weighing less that 2,950 pounds (average 2,612 pounds) by 100 pounds, or a 3.8% decrease, was associated with a 4.9% increase in fatality risk. The corresponding result for cars weighing 2,950 pounds or more (average 3,402 pounds), or a 2.9% decrease, was associated with a 3.2% increase in fatality risk. The Eqn 4-13 relationship can be considered a second law of two-car crashes.
Mass as a separate causal factor
Relationships so far introduced do not address how adding mass to an existing car affects risk. Nor do they answer the question, "Am I safer if I put bricks in my trunk?" Data sets rarely contain information on cargo, or on actual mass during crashes. All that is generally coded is a curb mass that is identical for all cars of the same make, model, and engine.
However, although FARS has no information on cargo, it does have information on the presence of passengers. By assuming that cars carrying a passenger were heavier by the mass of the passenger, the causal role of mass was estimated using head-on crashes between pairs of cars coded in 1975-1998 FARS.12 One car contained only a driver, while the other contained a driver and a right-front passenger. The effect of the passenger on driver fatality risk is shown in Fig. 4-11 (and listed also as the last entry in Table 4-2). The result is that when the curb masses of their cars are equal, then
4-14
That is, the accompanied driver is (14.5 ± 2.3)% less likely to die than the lone driver solely due to mass difference resulting from the passenger's presence. This result is a risk ratio. It therefore does not indicate the extent to which it reflects reduced risk to the accompanied driver and increased risk to the lone driver. To answer this requires a model.

Figure 4-11. How the additional mass of a passenger affects the probability that a driver is killed.12 FARS 1975-1998.

Model separating causal roles of mass and size
The previous two laws of two-car crashes, Eqns 4-3 and 4-13, can be combined to give
4-15 where r1,2 is the risk to the driver of car1 when it crashes into car2, assuming car1 and car2 have masses m1 and m2 and sizes equal to those of average cars of masses m1 and m2, respectively.12 The parameter t has the value = l/2 = 1.79 (where l is from Eqn 4-3). The masses in the intrinsic size term should be interpreted to mean sizes corresponding to cars with the indicated masses.
If cars of unequal mass crash into each other, the ratio of the risks to the drivers, r1,2/r2,1, is computed from Eqn 4-15 as
4-16
thus showing that Eqn 4-15 contains the first law, Eqn 4-3.
If the cars are of the same mass M, Eqn 4-15 computes the risk in each as k/M, the same as the second law relationship in Eqn 4-13. Thus Eqn 4-15 contains both laws of two-car crashes.
Computed versus observed effect of adding a passenger. If two 1,400 kg cars crash into each other, Eqn 4-15 shows each driver has identical risk r1,2 = r2,1 = 1 (taking k = 2,800 kg). This is the base case in Table 4-4, in which the mass of car2 remains fixed at 1,400 kg. If the mass of car1 is increased to 1,475 kg by adding 75 kg cargo, the risk to its driver is reduced to (1,400/1,475) = 0. 911 but the risk to the driver in car2 is increased to (1,475/1,400) = 1.098. Thus Eqn 4-15, which was derived only from the two laws, predicts that adding 75 kg leads to a value of R = 0.911/1.098 = 0.830. The closeness of this to the empirically observed R = 0.855 (Fig. 4-11) supports the validity of Eqn 4-15.

Table 4-4. Estimates from Eqn 4-15 of changes in risk when changes are made to an initial 1,400 kg car crashing head-on into another 1,400 kg car.

If the risks to the individual drivers are rescaled so that the risk ratio matches the empirically determined value, we conclude that the addition of a passenger reduces the risk to the accompanied driver by 7.5%, but increases the lone driver's risk by 8.1%. A small net risk increase of 0.3% averaged over both drivers results.
The risk reduction due to the presence of a passenger or other cargo is expected to apply also to single-car frontal crashes into objects that deform in ways not too differently from cars. The addition of cargo increases damage to the struck object, but with no corresponding increase in human harm. When the larger risk reduction from some single-car crashes is combined with the small net increase in two-car crashes, adding mass in the form of passengers reduces total driver deaths.
Different effect of replacing a car by a heavier one. If, instead of adding 75 kg of cargo, car1 is replaced by a different car that is 75 kg heavier and correspondingly larger, the driver of the 1,475 kg car will enjoy an 11.3% reduction in risk, but will increase the risk to the other driver by 6.9%. The net effect is a 2.2% net risk reduction averaged over both drivers. The maximum net reduction of 4.2% is produced when car1 has m1 = 1,670 kg. As m1 exceeds this value, the reduction in net risk declines.
When m1 = 2,015 kg there is no change in net risk. Replacing an m1 = 2,015 kg car by one even heavier leads to a net increase in risk in crashes with 1,400 kg cars. This can be understood in terms of the risk in the heavier car becoming so small that further proportionate reductions are of little consequence, while even small proportionate increases in the large risk in the smaller car add to total risk. Various crossover effects of this type have been observed - safety increases as vehicle mass increases, but not indefinitely.7
Replacing a 1,400 kg car with a heavier one will reduce total risk in crashes with 1,400 kg cars unless the replacement is more than 2,015 kg. Since only about 3% of cars in FARS are heavier than 2,015 kg, this hypothetical replacement would almost always reduce total risk. In general, replacing a car of any weight with a heavier car will in the vast majority of cases reduce total population risk. Eqn 4-15 always computes a net risk reduction when the mass of the lighter car is increased to become closer to the mass of the heavier car.
Equation expressing risks as functions of size and mass of both cars
As the first term in Eqn 4-15 relates to the car size, it is desirable that it should be expressed directly in terms of a linear dimension of the cars of indicated mass. While Eqn 4-12 relates mass to wheelbase, the same relationship will apply, to within a scaling constant, if we assume that vehicle length is proportional to wheelbase. It is convenient to define a risk of unity for a driver in a typical 1,400 kg car of length 4.8 m crashing into an identical typical car. This leads to
4-17

where L1 and L2 are the lengths of car1 and car2 (meters)
m1 and m2 are the masses of car1 and car2 (kilograms)
t = 1.79
c = 2 4.82.45 so that r1,2 = r2,1 = 1 when L1 = L2 = 4.8 meters and
m1 = m2 = 1,400 kg.
With the constant c specified, the equation measures absolute risks.
Making a car lighter and safer
The generality of Eqn 4-17 enables us to explore what happens to safety as characteristics of a car are changed. For case 1 in Table 4-5, car1 and car2 are both typical cars with length 4.8 m and mass 1,400 kg. This is the case that defines the unit of risk - so each driver has an absolute risk of one unit. In the other cases car2 remains unchanged but the properties of car1 vary. For case 2, m1 is increased to 1,475 kg without changing its size, thereby reproducing the same result as in Table 4-4. In case 3, the mass is unchanged but the length is increased by 20 cm. This reduces the risk to both drivers by 5%.

Table 4-5. Results for two-car crashes derived from Eqn 4-17. In case 1 the first car has the characteristics of a typical car, defined as m1 = 1,400 kilograms and L1 = 4.8 meters. In the other cases the characteristics of the first car are varied, but the second car is always a typical car.

In case 4 the mass of the first car has been reduced, thus increasing risk to its driver, and increasing overall risk. However, if this is accompanied by a 20 cm length increase (case 5), the driver in the lighter, larger car is now at reduced risk, while the other involved driver is at substantially reduced risk, for an overall net risk reduction. This is just one example of a combination of mass reductions and length increases that reduce risks to all. Through Eqn 4-17 different combinations of weight reductions and size increases that lead to safety improvements for all can be estimated. Additional examples are given in Ref. 25. Making vehicles lighter and larger requires use of more expensive lightweight materials. However, note that airbags cost US consumers $6.35 billion in 2003 (Table 12-6, p. 320), and making vehicles larger provides passive protection, while airbags do not (Chapters 12, 15).
Single-vehicle crashes
Single-vehicle crashes, which account for half of occupant fatalities, are conceptually simpler than multiple-vehicle crashes because outcome depends on the properties of only one vehicle. Other vehicles in the fleet are irrelevant. This same simplicity makes unavailable many of the methods used to study two-vehicle crashes, resulting in far less substantial knowledge about single-vehicle crashes.
Information is available from FARS on occupants killed in different types of vehicles in single-vehicle crashes. However, the same source provides no information for crashes in which no one is killed. In order to determine the crashworthiness in single-vehicle crashes of a set of vehicles, we need to know the number of people killed in the vehicles divided by the number of crashes in which the vehicles were involved. It turns out to be very difficult to discover how many vehicles are involved in single-vehicle crashes that do not produce serious injury or death.
There is a legal requirement to report a crash only when damage exceeds a specified monetary amount. Yet the cost of damage, and whether the vehicle can be driven after the crash, are all related to vehicle properties. For example, a heavier vehicle may uproot a tree and suffer little damage, whereas
a lighter vehicle might be more damaged if the tree remains standing. If a vehicle can still be driven after a single-vehicle crash, the driver may not wish
to inform the police, even if legally obliged to do so. Thus whether or not
identical crashes are reported depends on vehicle factors, but in ways that elude empirical examination.
Measures such as fatalities per police-reported crash, fatalities per injury, severe injuries per minor injury, injuries per police-reported crash are all ratios of crash outcomes. They are therefore subject to the same pitfalls mentioned in Interpreting risk ratios (p. 76). The absence of a dependence on mass in any such ratio does not mean that mass does not affect risk, but rather that mass has the same proportionate effect on the risks in the numerator and denominator.
The much-used measure, fatalities per million registered vehicles has the problems that different vehicles attract drivers with different use patterns and crash risks (Figs 4-1 through 4-4).
Pedestrian fatality exposure approach
Ideally, we would like to know the number of driver deaths from, say, impacts with trees divided by the number of impacts with trees. The FARS data provide information on drivers killed in vehicles striking trees, but little information on non-fatal tree impacts. However, if the vehicle strikes a pedestrian, this event is coded in FARS if the pedestrian is killed. Therefore, the ratio of the number of driver deaths to the number of pedestrian deaths for a set of vehicles is a surrogate for the number of driver deaths per tree impact, and accordingly measures how driver risk depends on the physical properties of the vehicle. This ratio plotted versus vehicle mass will therefore estimate how driver fatality risk depends on vehicle mass, subject to the additional assumption that the probability of pedestrian death is independent of vehicle mass. This is approximately so because even the lightest vehicle is so much heavier than the heaviest pedestrian (but see p. 73).
The finding that the ratio of driver deaths to pedestrian deaths is relatively independent of driver age supports the interpretation that the ratio reflects mainly the physical properties of vehicles.16, (p 73) Suppose two types of vehicles with equal crashworthiness are driven so that they differ by a factor of two in the number of crashes per year. The vehicles with the higher number of crashes would have twice as many driver fatalities from impacts with trees, but would also kill twice as many pedestrians, so that both vehicle types would have equal values of the ratio of driver to pedestrian fatalities.
Figure 4-12, derived from 1975-1983 FARS data,26(p 74) shows the number of driver deaths per pedestrian death versus car mass, which is interpreted to measure how driver fatality risk depends on mass. Fig. 4-12 shows a relatively noise-free relationship, with the data fitting well
4-18
where a is a scaling factor and b indicates the fractional change in risk per linear change in mass.
Systematic relationships between risk and vehicle mass are expected on physical grounds. Nearly all crashes are into objects that will to some extent move, bend, uproot, break, or distort, so that increased mass of the vehicle will systematically reduce the deceleration forces experienced within the vehicle. More damage being sustained by the struck object lowers risk to the vehicle occupants. While mass should not have a direct influence on rollover risk, vehicle size, which is correlated strongly with mass (Fig. 4-9, p. 78), does (as does, of course, the height of the center of gravity). Wider vehicles offer more resistance to rollover, and longer vehicles have higher lateral stability. Thus physical reasoning and the empirical data in Fig. 4-12 show that single-car fatality risk decreases systematically with increasing mass. Non-systematic behavior of measures based on driver fatalities per million vehicles likely reflects driver use and behavior effects.
It seems that the best way to estimate how single-vehicle (and rollover) fatality risk depend on mass is to assume that risk is a simple continuous function of mass, such as the functional form in Eqn 4-18. The slope parameter, b, is determined by the best fit to whatever data are available. While measures such as driver deaths per million registered vehicles have inadequacies, they are the best available data to determine the parameter. The value b = -0.00096 in Fig. 4-12 indicates a 9.6% decrease in single-vehicle fatality risk for an additional 100 kg of vehicle mass. This is equivalent to a 4.5% increase in fatality risk for a 100 pound reduction in vehicle mass, a measure we use because it has appeared in the literature.7 This is the first case in Table 4-6.
The results from Fig. 4-2 and Fig. 4-3 are from least-squares fits (poor fits) to the four points for each case. The estimate from Fig. 4-12 is expected to be high because it is based on assuming that pedestrian fatality risk does not increase with increasing mass of the striking vehicle, when in fact Newtonian mechanics indicates it does slightly (p. 74). Further indication that the risk of death to pedestrians increases somewhat with vehicle mass is provided by various results in Ref. 7. While there is substantial quantitative variation among the results in Table 4-6, all values of P are positive. This leaves little doubt that as the mass of cars or light trucks is reduced, fatality rates increase for single-vehicle crashes overall, and for the single-vehicle crash subcategories of rollover and crashes into fixed objects.
Corporate Average Fuel Economy (CAFE)
In response to the 1973 oil crisis, the US Congress passed the Energy Policy and Conservation Act of 1975, with the goal of reducing fuel use as a means of lessening US dependence on imported oil. The act established the Corporate Average Fuel Economy (CAFE) program, which required each vehicle manufacturer to meet a sales-weighted average fuel use standard for its passenger car and light-duty truck fleets sold in the US. Since 1996 the standards have been 27.5 miles per gallon for passenger cars and 20.7 mpg for light trucks (minivans, pickups, and sport utility vehicles).
A vehicle's fuel use is intrinsically related to its mass. The energy required to accelerate a vehicle from rest to a given speed is proportional to the mass of the vehicle. When in motion, rolling resistance forces are proportional to vehicle weight, which is proportional to mass. Thus the energy required to move a vehicle is linked to its mass through fundamental physical laws. Other factors being equal, making a vehicle lighter reduces its fuel use and increases injury risk to its occupants.
The influence of CAFE was more complex than merely leading to lighter cars and trucks. Indeed, the unintended consequences eclipse the intended. In order to meet CAFE requirements manufacturers had to sell a mix of vehicles different from the mix their customers wanted. This was achieved by, in effect, subsidizing small cars to increase their sales while adding a premium to the cost of large cars to discourage their purchase. Consumers who wanted larger vehicles but were reluctant to pay this premium found attractive alternatives in personal transportation vehicles classified as light trucks, which were subject to less stringent CAFE standards. Thus CAFE contributed to a major change in the types of vehicles on US roads.
Effect of CAFE on safety
From the time it was introduced, the effect of CAFE on safety was controversial. CAFE proponents seemed unwilling to accept that a policy they believed to be good for energy conservation and the environment could possibly cause additional deaths. They claimed that all cars becoming lighter would not increase fatalities because the driver in a two-car crash would benefit by being struck by a similarly lighter car. Such a claim flies in the face of two clear effects. First, when cars of the same mass crash into each other, risk increases as the common mass decreases (Fig. 4-10, p. 80), so that a fleet of identical lighter cars produces more two-car crash fatalities than a fleet of identical heavier cars. Second, about half of the car occupants killed are killed in single-car crashes in which risk increases with decreasing car mass. Drivers of lighter cars crashing into large trucks are at higher risk than drivers of heavier cars crashing into large trucks (p. 73 and Table 4-3, p. 76). Large trucks cannot be made much lighter as cargo is a major portion of their mass. There can be no doubt that policies that lead to a fleet of cars being replaced by a fleet of lighter cars must necessarily increase fatalities, because all resulting safety changes are in the same direction.
The widespread replacement of cars by light trucks complicates the safety computation. However, there is no evidence that this made the fleet safer by an amount that could negate the safety decreases from the lighter cars. Indeed, there is much opinion that these substitutions further reduced safety because of the higher rollover rates for SUVs, and suggestions that two-vehicle risks were increased by the presence of SUVs. The conclusion is inescapable that CAFE increased fatalities. A National Academies of Sciences report concluded "the downweighting and downsizing that occurred in the late 1970s and early 1980s, some of which was due to CAFE standards, probably resulted in an additional 1,300 to 2,600 traffic fatalities in 1993." (p ES-4)
The government body responsible for the CAFE program is the National Highway Traffic Safety Administration. Thus, the agency charged with reducing traffic deaths administers and supports policies that increase traffic deaths.
Effect of CAFE on fuel use
While there is no doubt that CAFE increased fatalities, its effect on fuel use is far less clear. CAFE unquestionably led to vehicles with higher fuel economy, meaning that a vehicle could travel further using the same amount of fuel. However, national fuel use depends on the numbers and types of vehicles in use and on how far they are driven, and not just on fuel economy.
When a vehicle buyer chooses a light truck rather than a large car, this choice increases the average fuel economy of the car fleet, and also increases the average fuel economy of the truck fleet. Although the choice increases the fuel economy of both fleets, it nonetheless reduces the average fuel economy of all vehicles. Another effect of CAFE is that it reduces travel costs per mile for most drivers. Reducing the cost of travel increases the amount of travel, with consequent increases in fatalities, and fuel use increases that partially offset any reductions from higher fuel economy.
Figure 4-13 shows the average distance traveled per vehicle per year since data were available. , The rate remained remarkably constant between 1947 and 1977, never varying outside the range 9,315 to 10,906 miles per vehicle per year. In the period after CAFE went into effect in 1978, the average increased substantially. As with all regulations that affect new vehicles, it takes about a decade before the vast majority of vehicles on the roads include the changes. This does not, of course, suggest that CAFE caused the increase in travel per vehicle that coincided with it. But it does show clearly the failure of a national policy aimed at reducing fuel for transportation, which had as its centerpiece CAFE regulations.27
Traveling 12,670 miles (the average for 2000) in a car meeting the CAFE standard of 27.5 miles per gallon requires 461 gallons of gasoline costing $691 at the typical cost in 2000 of $1.50 per gallon. The average annual cost of auto insurance in 2000 was $786. American families spend about three times
as much on eating out as on transportation fuel. One of the effects of CAFE
was to render fuel cost so inconsequential that it played no more than a minor role in vehicle or travel choices. Forcing consumers to drive higher fuel economy vehicles than they would have chosen reduced their incentives to carpool, use public transportation, more carefully plan shopping trips, live closer to work, etc.
While there may be some disagreement on the effect of CAFE on fuel use, there is no disagreement among economists that increasing the cost of a commodity reduces its consumption. Figure 4-13 shows clear drops in the distance traveled per vehicle related to increases in the cost and availability of fuel following the first and second oil embargoes of 1973 and 1979. An increase in the tax on fuel is guaranteed to reduce fuel use. This could be applied in ways that would avoid economic disruption, and, depending on the freely made spending choices of individuals, might increase or decrease safety.
The main reason one sees far more SUVs (and other large personal transportation vehicles) in the US than in Europe is not because Europeans do not like SUVs, but because European fuel costs discourage the selection of such vehicles. US consumers make rational choices based on US fuel costs. The US government used CAFE regulations as an excuse to avoid addressing policies that could really reduce oil imports. The absence of an effective policy precipitated momentous consequences for the nation and the world.
There is a taboo in US politics against even mentioning increases in the federal tax on fuel. The US approach to reducing foreign oil consumption is like a 300-pound patient asking a doctor how to lose weight, but insisting that the answer must not mention eating or exercise. If the one and only policy that can really affect energy consumption is off limits, then it would be preferable to formalize the decision to do nothing rather than enact policies which have only one clear effect - to increase fatalities.
Total safety, vehicle type, vehicle mass
The shift from cars to light trucks was not due entirely to CAFE - many consumers like such vehicles, especially SUVs. Quantifying how changes in the types and sizes of vehicles affect net safety is a problem of high complexity. Even if the fleet consisted only of cars, all driven identical distances in identical ways by identical drivers, the task of estimating overall effects from knowledge of outcomes for single- and multiple-vehicle crashes would not be trivial. The mix of single- and multiple-vehicle crashes is affected by rollover risk, which is related to car mass, thus making the mix of single-vehicle to multiple-vehicle crashes dependent on car mass. Cars of different masses are used in different ways, are driven differently, and attract different types of drivers. Even the same driver pursuing the same strategy may unknowingly drive cars of different mass in different ways (Fig. 8-1, p. 180).
When other vehicles are included, complexity and uncertainty increase. There is no analytical model of SUVs crashing into SUVs comparable to Eqn 4-17. What is more critical, there is no model of outcomes when cars and light trucks crash into each other. Simple risk ratios indicate that the car driver is 5 times as likely to die as the SUV driver when an SUV and a car crash into each other.17 When the comparison is restricted to vehicles of equal mass, the car driver is about twice as likely to die as the SUV driver.19 However, these are risk ratios and accordingly do not, by themselves, prove that a car-SUV crash poses more risk than a car-car crash. There are structural considerations that indicate that this is likely, but no quantification from field data.
The SUV, with a higher center of gravity than a car, offers less resistance to rollover (Fig. 3-12, p. 50). However, belt wearing in fatal rollover crashes is even lower for drivers of light trucks than for drivers of cars. This indicates higher risk-taking and law-violation by drivers of light-trucks, which would increase the overall fatality rates for these vehicles without regard to the properties of the vehicles.
Whether widespread replacement of cars by SUVs has increased or decreased the total number of US fatalities is difficult to answer. Cars and light trucks are driven different distances in different places by drivers with different characteristics. By far the most thorough study on this subject incorporated a host of confounding factors, including the age and gender of drivers of the different vehicle types, urban versus rural use, different speed limits, and night versus day.7 The distances driven by different vehicles were estimated from odometer readings in the NASS file (light trucks travel further than cars), and also by additional methods. The report, with over 300 pages, contains a wealth of information and insights relevant to many aspects of traffic safety. To compliment such completeness the author comments:
The analysis is not a "controlled experiment" but a cross-sectional look at the actual fatality rates of MY 1991-99 vehicles, from the heaviest to the lightest. Since most people are free to pick whatever car or light truck or van they wish (limited only by their budget constraints), owner characteristics and vehicle use patterns can and do vary by vehicle weight and type. This study tries, when possible, to quantify and adjust for characteristics such as age/gender or urban/rural, and at least to give an assessment of uncertainty associated with the less tangible characteristics such as "driver quality." But, ultimately, we can never be sure that a 30-year-old male operating a large LTV on an urban road at 2:00 p.m. in a Western State drives the same way as a 30-year-old male operating a smaller LTV/light car/heavy car on an urban road at 2:00 p.m. in a Western State. We can gauge the uncertainty in the results, but unlike some controlled experiments, there is not necessarily a single, "correct" way to estimate it.7(p 13)
The main findings were that decreasing the masses of cars or masses of the lighter categories of light trucks led to net increases in fatalities (fatalities to occupants of the vehicle plus fatalities to other road users). No clear difference in net fatalities resulted from decreasing the masses of the heaviest light trucks. Pick-up trucks and SUVs, had, on the average, higher fatality rates than MY 1996-99 passenger cars or minivans of comparable weight.
The finding that pick-up trucks and SUVs had higher fatality rates than cars of the same weight does not necessarily mean that a person switching from a car to an SUV would increase net fatality risk. A car would typically be replaced by an SUV of greater weight.
Another study including considerable detail confirms that driver factors, vehicle mass, and whether the vehicle is a car or a light truck have a clear influence on risk. Various studies have made claims that SUVs have produced dramatic increases in total deaths. Such studies have not taken into account, as is done in Ref. 7, the many factors that can influence results by large amounts. For example, the simple measure of deaths per million registered vehicles is elevated for light trucks because they are driven further, and in higher speed rural driving, than cars.
Does increase in number of SUVs increase risks to car drivers?
One common claim is that SUVs sharply increase total fatalities by increasing fatalities in the cars into which they crash. Such a possibility is inconsistent with Fig. 4-14. The number of car drivers killed in single-car crashes does not depend on other vehicles. From 1994 to 2002 the number of cars on US roads remained relatively constant while the total light truck population increased by more than 30%. , If the increase in SUVs led to large increases in fatality risk to car drivers from car-SUV crashes, the number of car drivers killed in two-vehicle crashes would increase relative to the number killed in single-car crashes, leading to an increasing trend in the percent of all fatally injured car drivers who were killed in two-vehicle crashes. No such trend occurred. Indeed, if there is a trend, it is in the opposite direction. The most plausible interpretation of the data in Fig. 4-14 is that SUVs posed about as high a risk
to car drivers as did the generally large cars they replaced. In any event, the
data are inconsistent with a large national fatality increase from SUVs killing large numbers of car drivers who would not have been killed if the SUV's had been cars.
What is effect of changing composition of fleet?
Unfortunately, complexity precludes a definitive conclusion. Overall, the evidence suggests that the widespread substitution of cars by SUVs may have increased net fatalities, but not by much. However, the uncertainties are so great that the effect could be in the opposite direction. For example, the SUV's may have siphoned off riskier drivers, thus increasing SUVs fatality rates and lowering car fatality rates.
The question of how the composition of the fleet affects safety is almost exclusively a question of changes in risk, given that crashes occur. In the aggregate, it is difficult to conclude even the direction of the effect. However, one can be confident that it is not one of the largest factors influencing safety. We noted previously that CAFE increased US fatalities by 1,300 to 2,600 in 1993, say about 2,000 per year. While of great importance, eliminating such an effect would reduce 2002 fatalities from 42,815 to 40,815. Important though this is, we show in Chapters 13 and 15 much larger and more clearly established effects due to non-vehicle factors.

Summary and conclusions (see printed text)

References for Chapter 4 - Numbers in [ ] refer to superscript references in book that do not correctly show in this html version.  To see how they appear in book see the pdf version of Chapter 1.