Take a case where you are asked to estimate the speed of a car that had struck a parked car. The striking vehicle is available for examination, however, the only damage information you have on the second vehicle is contained on an insurance photograph. The police took no measurements of the post-collision distances.

The following method that employs the same model that is used in the CRASH 3 program can provide you with what would normally be a minimum speed based on damage alone. Before we go any further it must be realized that if one is using energy based equations to solve for speed, one must account for all the energy. Some of the data that would provide us with the whole of the kinetic energy is missing.

The CRASH 3 model, which behaves like a linear dissipative spring with an offset, the relationship between impact force per unit width of residual crush is:

Where :

F = Crush force per inch of damage width required to produce the resulting crush deformation in lb/inch

A = Zero residual crush force. The maximum force per inch of damage width which will not cause permanent damage measured in lb/inch.

B = Slope of the force-crush curve The spring stiffness per inch of damage width measured in lb/inch²

G = A² / 2B The energy dissipated without permanent damage.

C = Residual crush (average depth)

At maximum crush assuming a force balance between the two vehicles (Newton’s Third Law):

Let A1 and B1 be the stiffness coefficients for vehicle 1 and A2 and B2 be the stiffness coefficients for Vehicle 2

A1 = 479.7 lb/inch A2 = 355.8 lb/inch

B1 = 52.2 lb/inch² B2 = 34.0 lb/inch²

C1 and C2 are the average crush depths of Vehicles 1 and 2 respectively. We are in possession of Vehicle 1 so determining C1 is simply a matter of measuring the crush.

C1 = 14 inches (the crush average across the width)

Which defines the value of the average crush on the second vehicle.

Also from the crush model we get the energy absorbed per unit width of the crush for vehicle 1 and the absent vehicle 2 as:

Substituting [Eq 3, 4 and 5] and simplifying we get:

The energy absorbed for the crush for the known vehicle (Vehicle 1) is obtained by intergrating the quantity E1 across the width of the crush (dw):

dw = 34.5 inches ( damage width of Vehicle 1)

From [Eq 6] and [Eq7] we get the energy absorbed by the crush on the missing vehicle:

where dw2 is the width of the damage on Vehicle 2 measured from the photograph

dw2 = 37 inches

Also known would be the weights of the two involved vehicles:

W1 = 3073.2 lbs

W2 = 4219.6 lbs

Velocities from kinetic energies would be added as all the kinetic energy came from Vehicle 1

So our first order estimate of absorbed energy produces an impact speed of 32.2 mph for Vehicle 1.

These formulae provide a starting point for speed estimates for frontal and rear end collisions when one of the vehicles is stationary.   In real world collisions one would expect post-collision movement and therefor the answer produced by these equations would generally be a MINIMUIM speed.

The author has examined these formulae against three head-on crashes conducted by CALSPAN for NHTSA with 1980s vehicles. The analysis of these three crashes produced remarkable results as follows:.

Case 414 which involved a bullet 1980 AMC Concord into a target 1980 VW Rabbit at a speed of 31.8 MPH. The analysis in the missing data formula produced a speed of 30.387 MPH. The result is within 4.5% of the known collision speed.

Case 447 which involved a bullet 1981 Chevrolet Citation into a target 1982 Volvo DL at a speed of 34.67 MPH. The analysis in the missing data formula produced a speed of 36.41 MPH. The result is within 5% of the known collision speed.

Case 810 which involved a bullet 1994 Dodge Omni into a target 1984 Chevrolet Celebrity at a speed of 37.7 MPH. The analysis in the missing data formula produced a speed of 37.31 MPH. The result is within 1.1% of the known collision speed.

If one checks the formula one also will notice that the weight of the target vehicle is not necessary for the calculations. Nor is the A crush coefficient for the target vehicle.