Pedestrian Vaults

Changing the Reference Frame

Copyright George M. Bonnett, JD 2005-2011 All Rights Reserved

The pedestrian vault: complicated and controversial – the perfect topic for examination. It sometimes seems that everyone involved with accident reconstruction has discovered a formula or formula variant computing pedestrian vault speed. There are at least a half-dozen computer programs that primarily deal with pedestrian and/or bicycle vaults or that have a section containing multiple pedestrian vault formulae.

What distinguishes a pedestrian vault from a regular vault and how do bicycles get included? In a regular vault the point of touchdown or landing can usually be determined. Within the field of accident investigation or reconstruction the vault formula is most often utilized with vaults involving launched vehicles that create easily discernible marks upon making contact with the struck surface. By contrast, a pedestrian vault is typically devoid of an easily discernible mark where the pedestrian made first contact with the surface. This undiscovered point of first contact problem also holds true for the operator of a bicycle or motorcycle launched by contact with a motor vehicle. It is the reason why pedestrian vaults must be separated from regular vaults and for the inclusion of the bicycle and motorcycle vaults.

The terminology or language involved with the investigation of pedestrian vaults contains some terms that are unique and specific to this type of vault analysis. One of these terms is “throw” distance. In a regular vault the distances involved in the analysis are the vertical and horizontal distances from the point of launch to the point of touchdown or first contact. As the point of first contact is not normally discernable when investigating pedestrian vaults, the horizontal distance is unavailable. Instead, the investigator uses the throw distance.

One common mistake the novice pedestrian vault investigator makes is assuming the throw distance begins at the point of first contact between the pedestrian and the vehicle. This is rarely the case. The throw distance is actually measured from the point of release or separation of the pedestrian from the vehicle. The other end-point for the throw distance is the point of final rest of the pedestrian and not where the pedestrian landed.

The throw distance, from release to final rest, covers not only the vault, but also the sliding distance of the pedestrian. This is the most distinguishing feature between the regular vault and the pedestrian vault – the pedestrian vault includes a slide over a surface to final rest. The throw distance covers the entire distance the pedestrian is thrown by the vehicle from release to final rest.

No wonder pedestrian vaults are so complicated. They involve both a vault over an undetermined distance and a slide over an undetermined distance. Please take note that the word undetermined was precisely chosen as while it may be immediately undetermined, it is not undeterminable.

Generally, vaults associated with accident reconstruction do not include corrections for air resistance, which is speed dependent. The masses are so large for the surface area of the pedestrian and the distances and speeds are so small, that the corrections fall within the margin of error relative to the measurement of the variables involved.

Another common mistake made by the uninitiated is assuming that there are different formulae for different types of vaults. Publications from some training institutions have broken the vault into falls, vaults where the landing is above the point of takeoff, vaults where the landing is below the point of takeoff, vaults where the launch angle is limited to a few degrees above or below the horizontal, and the list goes on. To set the record straight, there is only one fundamental, basic vault equation based on standard physics equations relating to distance, velocity and acceleration.

The analysis of a regular vault involves three primary variables that enable us to determine the forth variable, the vault speed – usually the unknown. These primary variables are the horizontal and vertical distances from launch to contact (as previously discussed), and the launch angle of the vaulting vehicle or object.

As a rule, the analysis of a slide or skid involves a beginning speed, a final speed, a distance and the friction factor between the surface and the object. If three of the variables are known, the forth variable or unknown is easily solved using a relatively simple non-linear equation.

The vault and the slide are the two components of the pedestrian vault. The unique feature of the pedestrian vault is that the throw distance is shared in an initially undetermined ratio. As is all too often true with the regular vault, the launch angle of the pedestrian vault is probably unknown.

Eating an Elephant

So how does one eat an elephant? One eats an elephant one bite at a time. If that works for elephants, it may also work for pedestrian vaults.

Pedestrian vaults deal with the pedestrian from the release or separation from the vehicle to the point of final rest, the throw distance. This throw distance involves both a vault and a slide. Since this is the essence of the pedestrian vault, why is this not the primary focus of the investigation into a pedestrian vault? Once separation occurs, the vehicle no longer has any influence on the pedestrian unless there is a secondary collision. The resulting motion of the pedestrian as governed by the laws of physics is the primary focus of our investigation.

The 600 Pound Gorilla

Current methodologies appear to examine the vehicle-pedestrian geometry in assuming a non-reported launch angle to solve for the vault speed. The subject system method examines the vault-slide relationship for the given throw distance using a narrow range of appropriate input launch angles in determining the speed. The expert can then examine the effect and determine the validity of all of the inputs, including the launch angle.

For any given launch angle and friction coefficient between the pedestrian and the surface over which the pedestrian is sliding, the vault and the slide components of the pedestrian vault are inexorably tied to each other. This tells us that if the launch angle and surface friction are known for any given throw distance, then solving for the vault speed of the pedestrian is simply a matter of doing the mathematics.

If the throw distance is kept as a constant, then the only way the eighty plus different currently published pedestrian formulae can arrive at different solutions is by varying either the launch angle or the friction factor of the pedestrian. As every pedestrian vault formula requires an input of the friction factor, the only remaining variable is the launch angle. Because of this fundamental relationship based on physics principles, the only real difference between all of the existing pedestrian formulae is the assumed launch angle.

Almost all of the pedestrian formulae are designed to handle one of two different types of problems: 1) the wrap, or 2) the frontal projection.

The above statement is technically correct, but when examined closely, all that is at issue is the launch angle. Generally a frontal projection will have a launch angle less than or slightly above zero. Launch angles much greater than zero are usually the result of a wrap. Again, it is simply a matter of the assumption made for the launch angle that is the difference in these two main formulae types.

The Vault

How can we arrive at a solution for a known launch angle, throw distance, elevation change and friction coefficient?

There is almost enough information to use the regular vault formula to solve for a vault speed. Using the information at hand, a vault solution is possible but there is no distance remaining to dissipate the energy as a result of the horizontal speed of the pedestrian. If the entire throw distance is used for the vault, there is no linear distance for the slide. The pedestrian would have landed at the final rest position with all of the horizontal speed still intact.

How is the percentage of the throw distance allotted to the vault portion of the combined maneuver determined? At this point the answer should be crystal clear. We have only two choices. We either quit, or we guess.

The Guess?

The only option left is to assume a horizontal distance for the vault. With this assumption, a speed can be computed using the vault formula.

The Skid

With the vault velocity and the known launch angle, trigonometry provides a horizontal velocity for the pedestrian. The horizontal velocity of the vault is the initial speed in the skid. The final speed is obviously zero. With a known initial and final speed and a deceleration factor, a simple computation solves for the distance required for the change of speed.

Eureka!

When the distance computed for the skid combined with the initial assumed vault distance is equal to the throw distance, the only solution for the given launch angle is a reality. A simple iterative process generates the required solution.

So where does this leave us?

If this reference frame is utilized when investigating the pedestrian vault, in lieu of a plethora of formulae all generating disparate solutions based on the same data, the investigator can use the regular vault formula and the distance formula for a deceleration to arrive at a solution that is demonstrably correct using scientific principles. The formulae involved are readily available in any college level basic physics text.

This does not denigrate any of the studies done in an effort to validate the other published pedestrian formulae. These studies can be used in conjunction with the iterative process described above to validate the process.

Most of the published pedestrian formulae do not require a launch angle. This angle will have to be a supposition supplied by the individual investigator. While it requires an assumption on the part of the investigator, it is the investigator who is in the best position to determine the angle or range of angles that are most suitable for use in connection with the specific incident under investigation.

This process does have one nasty drawback. The iterative process is time consuming unless the investigator is extremely lucky or has access to a computer program that will do the iterations.

Below, REC-TEC Professional software will be used to illustrate the principles that have been described. The results will be compared to several of the more universally accepted pedestrian formula.

Example 1 (Forward projection)

Throw distance = 65 feet Vertical distance = –2.5 feet

Coefficient of friction = .7 Launch Angle = Zero degrees

Common Pedestrian Vault Formulae

The computed solutions are within a narrow band ranging from a low of 28.53 (IPTM general purpose formula) to a high of 36.9 with three of the solutions at 31.3453. Collins suggests the correct friction factor is .8 and a solution is shown for both the .7 and .8 values.

Below is a solution obtained using the system described earlier. It took the software 38 iterations for the solution using a sophisticated algorithm in which it carried out the comparison to 9 decimal places.

This system also generated the vault distance (18.12 ft) and the slide distance (46.88 ft). As a result of this breakdown for the distances, the times for the vault and slide are also available. The diagram shows a scale vault and slide.

The program can also generate the friction value required for the slide in order to meet the throw distance value for a specific speed. As a self-check on the software algorithm employed, the computed value of 31.3453 will be entered as the vault speed. This should compute to a value very close to the .7 entered as the friction value in the original problem.

The speeds for the IPTM, Collins (.8) and Searle formulae will also be entered and shown on the following pages. While the IPTM formula is a general pedestrian vault formula, the NUTI and Collins formulae are frontal projection formulae. Searle’s formulae allow for an initial high acceleration at first contact and appear to be general purpose, but that determination will be left to the reader.

Algorithm Check Value: : Friction Required = 0.7

IPTM Computed Speed : Friction Required = 0.5605

Collins (.8) Computed Speed : Friction Required = 0.8

Searle (Minimum) Computed Speed : Friction Required = 0.6425

Searle (Maximum) Computed Speed : Friction Required = 1.0419

Computation for Optimum Angle with lowest vault speed consistent with other data

The image above shows the absolute minimum speed required to complete the maneuver with a .7 coefficient of friction over the throw distance of 65 feet. Any lower vault speed would result in the pedestrian not being able to cover the entire distance at the given friction level.

The speeds for the IPTM and Searle (Minimum) are not really problematic. The IPTM formula is actually a general-purpose formula not restricted to zero degree frontal projections. The Searle (Minimum) formula falls within the range established by the IPTM formula as it relates to the NUTI and Collins formulae.

A pedestrian vault with a 10-degree launch angle would include the lower end of the spectrum for wrap projections. The pedestrian is launched at an upward angle to the horizontal as a result of the geometry of the frontal area of the striking vehicle and the pedestrian’s center of gravity location. The image below shows a launch angle of 10 degrees with the other input data unchanged. The computed speed for this maneuver is 28.36 miles per hour, very similar to the IPTM computed speed (28.53) for a general-purpose launch.

Example 2 - (Example 1 with a 10-degree Launch Angle)

The actual IPTM, NUTI, Collins and Searle formulae are shown on page 5. Note that none of them use a launch angle as part of the computation except as is inherent in the change of vertical height to the point of landing.

Looking at the pedestrian vault from the vault/slide perspective has certain advantages including the ability to separate the throw distance into its component parts. This breakdown may point to areas of the scene deserving increased scrutiny in a search for corroborative evidence. It will also point out the sensitivity of the vault speed to the various input variables.

Disregarding the slide component of the throw distance has a dramatic effect on the vault speed as evidenced by the image below. The airborne vault equation should be used only if a point of first contact with the surface can be determined. In this situation the horizontal distance from release to point of touchdown does not constitute a throw distance. Throw distance by definition must have a sliding component with no clear demarcation between the vault and the slide.

A system capable of utilizing launch angles provides the advantage of iterating the launch angle to test the sensitivity of the vault speed. Testing the sensitivity of the vault speed to changes in input data is a valuable tool in the analysis of the pedestrian vault. Analysis using a single systematic approach should yield more useful information than attacking the problem with the number of pedestrian vault formulae currently available to the investigator.

There are an estimated 80 plus formulae competing for prominence in the reconstruction community. All of them must deal with the friction of the pedestrian, the vertical height of the center of mass of the pedestrian and the throw distance involved. The wrap formulae strive to generate vault speeds by imputing launch angles based on geometry and other factors including the friction between the vehicle and the pedestrian. The specific data required for these formulae are rarely available.

Iteration/Finite Difference Analysis Menu

Iteration of Throw Distance

Iteration of Vertical Distance

Iteration of Pedestrian Friction Value

Iteration of Takeoff (Launch) Angle

Iteration of Pedestrian Vault Speed to solve for Friction Value

Finite Difference Analysis

Graphics of Finite Difference Analysis Values

The iteration tables on the previous pages show the sensitivity of the vault-slide integration system to the changing values for all of the individual input variables. This iteration approach is a useful tool when dealing with a small number of formulae or a systematic approach but becomes un-wieldy when dealing with a large number of diverse formulae.

One of the real strengths of the vault-slide integration system however is that it is possible to utilize Finite Difference Analysis to test the sensitivity of the system to changing values of a particular variable. In the process it generates the statistical uncertainty of the system for the specific ranges assigned to the variables.

Example 3

It is now time to methodically plow through the step-by-step process of arriving at a solution for a pedestrian vault. Manually calculating a solution will give us a better understanding of how the system works.

Throw distance = 125 feet Vertical distance = –2.5 feet

Coefficient of friction = .8 Takeoff Angle = 7.5 degrees

As previously discussed, the throw distance needs to be divided into the vault distance and the skid distance.

There is an old saying amongst homicide detectives that it doesn’t matter what a suspect says, as long as you can get them to say something. The rationale is that if the suspect is telling the truth, it should be provable and if he/she is lying that should also be provable. Either way, the statement will go a long way in either eliminating or convicting the suspect.

The same rationale is used in solving this problem. Since it is difficult to interrogate either the deceased pedestrian or the throw distance as to the exact vault distance, a vault distance will be chosen at random. The initial vault distance chosen is 65 feet.

The standard vault formula:

V_v = Sqr (g / 2 * X² / cos²(A) * (tan(A) –Y))

V_v = Sqr (32.2 / 2 * 65² / cos²(7.5) * (tan(7.5) –-2.5))

V_v = 79.1099 feet per second

Now we have enough to confirm or eliminate our suspect – the 65-foot vault distance. Using this information the distance of the slide can be determined. Only a really retarded rookie would subtract the 65 feet from the 125-foot throw distance and say the slide was 60 feet. The expert calculates the distance for the slide from the known initial speed

The next step in the process is to determine the horizontal speed of the pedestrian.

V_h = V_v * cos(A)

V_h = 79.1099 * cos(7.5)

V_h = 78.4331 feet per second

With a horizontal speed and a friction (deceleration) factor, a time for the slide is computed. The distance for the slide is then computed using the time.

T = (V_i – V_f) / (f / g)

T = (78.4331 – 0) / (.8 / 32.2)

T = 3.0447 seconds

D = V_i * T – f * g * T² / 2

D = 78.4331 * 3.0447 – .8 * 32.2 * 3.0447² / 2

D = 119.4051 feet

Throw Distance = 119.4051 + 65 = 184.4051 feet

It looks like 65 feet gets a pass on this one, as it obviously does not provide a solution (125 feet) to the problem. However, all is not lost, as now we know that the suspect vault distance must be less than 65 feet.

The next suspect distance is 50 feet. The same procedure is used for 50 feet as was used for 65 feet. While this suspect turns out much closer, it is still too long to be correct.

The third suspect is distance 45 feet. Again the same procedure is used, but this time the suspect is too short. This is the process that will continue to be used until arriving at a perfect result of 125 feet when the vault distance is added to the slide distance.

When using this methodology, and doing the calculations manually, humans have an inherently superior ability over the computer. Human reason and logic team up to help choose the next suspected distance. The computer can use only the pre-programmed algorithm in search of the solution.

Modern computers can arrive at the solution in less time that the human heart muscle takes to contract, but it does have to take more steps than we do. Hopefully that thought will give us some solace. It is analogous to the race between the hare and the tortoise.

Computer Solution to Example 3

Expanded Graphics for Example 3

Hypothetical 2-Sigma (95%) Confidence Level for each Variable

Result is Speed (43.723 m/h) with Uncertainty (+/- 3.5663 m/h) at 95% Confidence Level

Graphical Representation of Finite Difference Analysis high and low variable values

Caveat

Searle’s theory of increased acceleration at touchdown has merit. However, the fall distance from the apogee may not be as high as is first assumed and there may be some bounce. The determination of how, or if, this is analogous to a skip-skid will be reserved for the expert in the particular case. In any event, it is something the well-prepared expert should be ready to discuss in presenting his opinion.

Assumptions and the Courts

The courts, especially in the federal system, do not like assumptions. Many have to be tolerated but they are usually limited to cases where the assumption is then either proved or disproved. Almost all of the current in-vogue pedestrian vault formulae make an assumption as to the launch angle of the pedestrian.

Courts in the federal system have rigorously excluded software that makes assumptions in the computations that are not reported. Assumptions that are reported that can then be examined as to their effect on the outcome are permitted. Assumptions by computer programs that are surreptitious in nature and not reported, and therefore cannot be examined, are routinely excluded from evidence.

As most of the current pedestrian vault formulae are doing exactly the same thing, one could certainly argue for their exclusion.

The Thrill of Victory and the Agony of Defeat

It is always comforting for an expert to go into deposition or trial knowing that the common formulae upon which their testimony is based can be found in any physics book in the world. The certain knowledge that underlying the opinions to be expressed are the same principles of basic physics that have been universally accepted for hundreds of years simultaneously creates both tranquility and excitement.

On the other hand, the thought of having to explain the self-serving selection of only a handful of formulae from a vast pool of over eighty must create an intense feeling of anxiety, especially when these exotic formulae were used to explain the ballistic and sliding motion of a struck object. In addition to the justification of the individual selections, the expert certainly should be prepared to discuss and differentiate the attributes and oft-times disparate solutions generated by both the selected and non-selected formulae – a daunting task.

Copyright George M. Bonnett, JD 2005-2011 All Rights Reserved