There are several pieces of evidence that can be used in estimating the speed of a vehicle involved in an accident. Skidmarks and the amount of damage (or crush) to the vehicle(s) are those most commonly used. Consider the case of a weary driver who went to sleep while approaching a curve on a rural two-lane road at night. He evidently awoke to find himself on a gravel shoulder headed directly toward a concrete retaining wall. Maximum braking was applied while on the gravel, traveling 8 ft on loose gravel and 32 ft on grass on a fairly level surface before hitting the wall head-on. The work done by friction indicates the change in kinetic energy (KE) prior to hitting the wall, but how do we determine the total change in KE?
The retaining wall had only minor surface damage. In addition to braking, most of the initial KE was transformed into strain energy through crushing the front of the car. Experience says that perhaps speed can be determined from the amount of crush. Automobiles are very complex structures composed of sheet metal, frame, engine, transmission, axles, and other metal, plastic, glass and fiberglass parts. It is practically impossible to determine the exact relationship between the amount of crush and the impact speed. Also, crush can be very difficult to quantify. Crush is represented by a complex three-dimensional surface, and is never uniform in any direction. However, simplified models and procedures have been developed that allow reliable estimates of the relationship between KE & crush. If the KE after impact and the KE consumed during impact is known, the KE just prior to impact can be determined, and from this the speed at impact.
One model assumes that the vehicle behaves like a linear plastic spring. This gives a linear relationship between impact force and crush depth. Thus, a relationship between impact speed V (mph) and D (average crush, inches) can be obtained,
V = V0 + k × D
V0 is the maximum speed at which no crush occurs (mainly due to bumper structures), and k is the crush stiffness for the particular vehicle. The National Highway Traffic Safety Administration (NHTSA) and other organizations conduct crash tests and publish data from which values for k may be calculated. The NHTSA data is available on its website (www.nhtsa.gov-under crash data/research). Usually, data can be found for a particular vehicle or one that is similar. If not, it might be necessary to conduct an independent crash test to determine "k" for that particular model. Of course, the linear model is a simple approximation, and alternative models have been developed to overcome its limitations. One model assumes a car is like a reinforced thin walled column similar to a reinforced can. This model behaves elastically for small deflections, but buckles like a crushed can above a critical impact level.
The case above involves a 2000 Toyota Camry weighing 3596 pounds loaded, and the crush depth was calculated to be 22.4 inches. Using the linear model, NHTSA data indicates a crush stiffness of 1.6 mph/in. Using V0 = 2.5 mph, the speed at impact is calculated to be 38 mph. At high speeds, we can assume that the impact is plastic and the KE after impact is zero. Using coefficients of friction of 0.4 for the gravel shoulder, and 0.25 for the grass, the work done by friction prior to impact is 40,275 ft-lbs. Adding this to the KE at impact gives an initial KE of 216,839 ft-lbs, which translates into an initial speed of 42.5 mph.