Turns, including lane changes, present some interesting and unique problems for the accident reconstructionist.  Not only are the maneuvers themselves complex, but they are modeled by several different equations, each giving a different answer.  In addition, there are two distinct methodologies making it seem as though we are speaking different languages when it comes to the terminology used to describe lane changes.

 Unless the turn involves a critical speed maneuver, which would use all of the available lateral friction to change the direction of the vehicle, the turn is usually described by the lateral (Y-axis) acceleration factor for the maneuver.

 If the vehicle is initially traveling on the X-axis, how far must it travel in the X direction while it moves a given distance on the Y-axis?  This Y-axis distance is usually referred to as the lateral distance.  At first blush, it would seem simple enough.  With a given speed over a surface with a known coefficient of friction, the process attempts to determine how far the vehicle must travel on the X-axis to traverse a given lateral distance that is perpendicular to the original direction of travel.

 The mathematical models generally assume that the maneuver is accomplished at a constant speed and that each turn is a perfect arc with instantaneous changes of direction. The models disregard the reaction time necessary for steering input as well as the mechanical lag time for the steering and tires.  Obviously this is an impossible maneuver.  The formulae do not take these lag times into consideration, as they are variable from car to car, turn to turn, and driver to driver.  The turns are treated as simple arcs by the formulae, and this causes a problem when these unmodified formulae are taken from the desk to the street.

 In developing the formulae for displacement on the X-axis when dealing with an arc, the chord and middle ordinate are the weapons of choice.  When brought to the roadway, the terminology changes to lateral acceleration and lateral displacement for the turn(s) involved in the maneuver.  While some may consider this distinction too subtle to warrant attention, nothing could be further from the truth.

 Before delving into the chaos resulting from the differences in terminology, it is necessary to articulate a few ground rules.  If the maneuver is not a maximum rate change of direction, also known as either a critical speed swerve or yaw, then there is probably no need to decelerate during the course of the maneuver.  In either event, determining the distance involved without decelerating in the course of the maneuver is an excellent place to start.  The physical and mechanical reaction times involved in getting the vehicle to change directions can be handled by some future modification to the basic model.

There are six (6) sets of formulae that will be examined.  Each of the sets has two formulae, one for a turn-away and one for a lane change.  The formulae will be shown in their simplest form, dealing only with the turning maneuver, and will not include the time from perception to turn.

In order to evaluate the equations, the same data will be used for comparison purposes.  All of the equations use basically the same information. The variables include Lateral distance, Acceleration factor on the Y-axis, and Initial speed. 

For simplicity, the problems will be worked in the Imperial System using the following values:

            L          =          Lateral distance                                             =          12 feet

            f_Ay       =          Lateral acceleration factor on the Y-axis     =          0.2

            V          =          Initial Speed in feet per second                    =          88 f/s

            g           =          Acceleration due to gravity                           =          32.2 f/s/s

            Sqr ( )   =          Square root.

 

Formulae:

1.      Limpert

        A. Turn-away Distance        =  0.235 * V * Sqr (L / f_Ay)                 =          160.1866 ft

        B. Lane change Distance     =  0.47 * V * Sqr (L / f_Ay)                   =          320.3731 ft

2.      Lofgren

        A. Turn-away Distance       =  0.225 * V * Sqr (L / f_Ay)                  =          153.3701 ft

        B. Lane change Distance     =  0.45 * V * Sqr (L / f_Ay)                    =          306.7402 ft

 3.      Daily

        A. Turn-away Distance       =  0.249 * V * Sqr (L / f_Ay)                  =          169.7296 ft

        B. Lane change Distance     =  0.499 * V * Sqr (L / f_Ay)                  =          340.1408 ft

 4.      Mitchell

        A. Turn-away Distance       =  V * Sqr (L / (0.5 * g * f_Ay))              =          169.8812 ft

        B. Lane change Distance     =  2 * V * Sqr (L / (0.5 * g * f_Ay))        =          339.7624 ft

 5.      Bonnett

        A. Turn-away Distance       =  V * Sqr (2 * L / (g * f_Ay))                 =          169.8812 ft

        B. Lane change Distance     =  2 * V * Sqr (L / (g * f_Ay))                 =          240.2483 ft

6.      Mocklin

        A. Turn-away Distance       =  Sqr (2 * L * V² / (g * f_Ay) - L²)         =          169.4568 ft

        B. Lane change Distance     =  2 * Sqr (L * V² / (g * f_Ay) - (L/2)²)   =          239.9484 ft

 

The only agreement is between 4A and 5A with 3A and 6A very close and (almost) within the round off error caused by using a square root constant for some of the terms.

A striking similarity exists among the answers of sets 2 through 4.  In each instance the lane change is twice the turn-away distance.  The exceptions are 5B by Bonnett and 6B by Mocklin.

  

A Closer look

The lateral distance is 12 feet.  It takes 1.9305 seconds to travel 12 feet when accelerating at a .2 acceleration factor.  This now becomes the time frame for the turn-away maneuver, not only for travel along the Y-axis, but also along the X-axis.  In order to calculate the distance along the X-axis for a turn-away, it is necessary to multiply the X-axis speed by the time frame established for the Y-axis acceleration, which is 1.9305 seconds. This results in an X-axis distance of 169.8812 feet.

The difference in the fundamental methodology seems to be centered on the relationship between lane change and lateral distance.  The term lane change normally refers to a vehicle moving from one lane to another.  The distance between the centers of the respective lanes would be the lateral distance involved in the lane change.  In a lane change the vehicle returns to a path that is parallel to the original path of travel, unlike a turn-away maneuver where the vehicle is no longer headed in the original direction, but at some angle to the original direction.

When using the methodology in formulae sets 1 through 4 above for a lane change with a lateral distance between lane centers of 12 feet, as in the exemplar problem, there is something drastically wrong as evidenced by the difference between those solutions and the lane change solution from set 5.

When confronted with an obstacle that requires a lateral movement of the center of mass of the vehicle of 12 feet laterally, there are two basic turning options available to the driver.  The turn-away, which avoids the obstacle but may cause the vehicle to leave the roadway during the recovery, and the lane change or S turn, that places the vehicle in a path parallel to the original path as the object is safely passed.   

 

 

Given the same lateral distance for the lane change as for the turn-away, the problem will manifest itself.  It is improper to use the same lateral distance without changing the methodology as used in formulae sets 1 through 4.  The lane change involves a change in the direction of the acceleration at the mid point of the lane change. It is an “S” turn that is a parabolic curve or a segment of a parabola.  It involves two accelerations in opposite directions.

 

 

 

 

Using the given data, the lateral distance is 12 feet (center to center of the lanes) for the lane change. The lane change requires an instantaneous reversal of direction at the mid point of the total lateral distance.

  

  

T  = (-Vi + Sqr (Vi² + 2 * D * fa * g)) / (fa * g)

This formula is used to compute the time of acceleration for a known distance with the term fa representing the acceleration factor.  When used to calculate the time for the acceleration in the lateral direction, the initial velocity is zero and the equation can be simplified to T  = Sqr (2 * D * fa * g) / (fa * g). 

 T = Sqr (2 * D  / (fa * g))                                                       Canceling

 T = Sqr (2 * L / (f_Ay * g))                                                       Substituting

 The acceleration from zero over a distance of 6 feet takes 1.365 seconds and reaches a speed of 8.7906 f/s using a .2 lateral acceleration factor.  At the mid-point, the vehicle decelerates from the lateral speed of 8.7906 f/s to a lateral speed of zero, which takes another 1.365 seconds.  Remember, this is a lane change. The vehicle is moving laterally from one lane to the other. 

Multiplying 88 by 1.365 we get 120.1242 feet to the reversal point of the turn and multiplying that by 2 gives 240.2483 feet for a lane change with the center of mass of the vehicle moving 12 feet laterally.

  

How Close is Close Enough?

The foregoing analysis is not perfect for several reasons: 

·        The analysis makes three critical assumptions:

o       All changes of direction are instantaneous.

o       The vehicle maintains a constant speed throughout the maneuver. 

o       The acceleration model is a parabolic curve.

·        The use of lateral acceleration creates a parabolic curve and not a true arc of a circle.  The resulting time frame for the lateral acceleration is flawed as a result of the difference in the curves.  This flawed time frame results in an error in the X-axis distance traveled in the course of the maneuver.

·        The vehicle does not travel along the X-axis; it travels along the arc or arcs depending on the maneuver.  The actual path distance is somewhat longer than the X-axis distance, 240.721 feet versus 240.2483 feet for the example lane change problem.  How much longer depends on the lateral distance traveled. 

·        There is a slightly longer analysis that is extremely accurate. Since these formulae do not account for lag times in direction changes resulting from either the steering input or the mechanical lag time of the tires themselves, the enhanced accuracy of the accurate analysis, according to many, is hardly worth the effort in the majority of reconstructions.

 

A Better Mousetrap

The formula for centripetal acceleration found in any physics book is a_c = DV² / R.

Solve for R and substitute f*g for acceleration              R = V² / (f*g)

One commonly used formula for Radius is                   R  = (C² / (8*L)) + (L/2)

where L = middle ordinate and C = Chord 

Solve for C² and substitute V² / (f*g) for R                  C² = 8*L*V² / (f*g) – 4*L²

Substitute 2*X-axis distance (2*D_x) for chord              (2*D_x)² = (8*L* V² / (f*g) – 4*L²)

(Full chord is 2 times X-axis distance)

Factor and reduce                                                        D_x = Sqr (V² / (f*g)*2*L – L²)

Distance to swerve  (turn)                                         D_x = Sqr (V² / (f*g)*2*L – L²)

Distance to swerve & return to parallel                   D_x = 2*Sqr (V² / (f*g)*L – (L/2)²)

 

Lagniappe:  A side-by-side comparison of the formulae below reveals that the distance slide to stop is one-half of the radius of a yaw for the same friction surface and velocity.

 

Radius of a circle                                                       R   = V² / (f*g)

Distance slide to stop                                                D_ss = V_(initial)² / (2*f*g)

 

Which Witch is which?

It would appear that the methodology taken by Bonnett and Mocklin for the lane change lateral distance is correct.  Additionally, these two formulae sets complement each other with one offering absolute accuracy and the other offering speed and increased versatility.  The increased versatility comes from the time computation for the lateral acceleration.  If used in conjunction with a change of speed of the vehicle, as in passing or braking, a time reference can easily be established that is unavailable using the more accurate set of formulae.

So how did all these others get it so wrong?  They didn’t.  Remember the reference to chaos caused by the difference in terminology?  The formulae in sets 1 - 4 do not really have lateral distance as the input variable.  While some even state that they use the lateral distance, it is not the lateral distance between lane centers.  Formulae sets 1 - 4 require the middle ordinate of the arc, not the distance between the lane centers.  Unfortunately, in many of the “formulae” books on the market, this distinction is not made clear and the user is left to proceed at their peril.

 

A Different Road

The formula for the path taken by the vehicle is D_path = R * p / 180 * (sine^-1 (D_x / R)).  This formula uses the X-axis distance and is, therefore, correct for both the swerve and the lane change calculation.

 

A Different Look

The turn and lane change formulae presented thus far all solve for the X-Axis distance.  In order to determine if an evasive turn maneuver is even possible, a different approach is necessary. 

For a given Distance on the X-Axis, the maximum Y-Axis (lateral distance) that can be negotiated is found by using L_critical = Radius - Sqr (Radius² – D_x²), assuming a constant speed for the vehicle.

For a given Distance on the X-Axis, the formula for the maximum Y-Axis (lateral distance) that can be negotiated during a lane change maneuver, assuming a constant speed for the vehicle, is L_critical = 2*(Radius - Sqr (Radius² – (D_x / 2)²)).

If the lateral distance exceeds L_critical, braking will reduce the impact speed.  As the percent of braking increases, the radius of the turn will also increase until all of the available friction is used in the deceleration.  With no friction available for the turn, the vehicle will travel in a straight line.

If a turn is attempted when the object extends beyond L_critical, a collision will occur at the (constant) initial speed albeit the contact angle will have changed which may reduce the severity of the collision. 

 

Supplement to the Basic Formulae

The following supplemental calculation can be added to the value obtained using any of the formulae sets, which will add the distance traveled during the reaction time between perception and the start of the maneuver:

• + V * T_r                      

Where T_r is the reaction time between perception and the start of the maneuver.

 

 

Copyright George M. Bonnett, JD  2001  All rights reserved

 

Special Acknowledgment

 Mocklin, Ross J.J.:  Ross introduced me to this basic concept of accurately dealing with the problem of turns.  Ross was my partner at New Orleans Police Department, and became my closest friend long before he became my first accident investigation instructor.  Ross and John Rigol, Jr., a lieutenant on the Louisiana State Police and a wiz at mathematics, introduced me to accident investigation in 1982.  John and I have both contributed to the development of the formulae now presented, but the concept has not changed, nor have the results to the last decimal place.

  

Bibliography

Daily, John: Fundamentals of Traffic Accident Reconstruction, IPTM, Jacksonville, Florida 1989

Limpert, Rudolf: Motor Vehicle Accident Reconstruction and Cause Analysis (Sixth Edition), The Michie Company, Charlottesville, Virginia, 2001

Lofgren, Myron J. Handbook for the Accident Reconstructionist, IPTM, Jacksonville, Florida 1983

Mitchell, J.F.: International Guidebook for Traffic Accident Reconstruction (Second Edition), J.F. Mitchell, Barrie, Ontario, Canada 1999

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Last edited on Thursday, 22 September 2005 04:25:27 PM -0400