R
E C - T E C

ACCIDENT .
. . . . RECONSTRUCTION
. . . . . .
SOFTWARE

*"**when**
**performance**
**counts"*

**EFFECT OF GRADE ON KINETIC FRICTION IN ACCIDENT
RECONSTRUCTION**** **

**Copyright (C) George M
Bonnett, J.D. 1992**

**All rights reserved.**

**ABSTRACT**

This paper details a method of adjusting the drag factor of a vehicle when there is a difference of grade between the tested surface and the actual accident surface. This method is based on the sum of the forces acting on the object and is accurate for any grade. The popular method used by most accident reconstructionists consists of adding the signed difference in grade to the test drag factor. Many reconstructionists believe that the adjusted drag factor obtained by this second method is relatively accurate only for grades of ten percent (.1) or less.

**THE RECONSTRUCTION OF AN ACCIDENT **often
involves taking measurements that are difficult or dangerously
impractical for the investigator. One of these measurements is
the drag factor of the vehicle (or object in question) at the
scene of the accident. Often the actual vehicle is not available
for the tests and it is not always possible to perform the tests
at the scene of the accident. There may be times when the slope
of the roadway makes the testing unreasonably dangerous, placing
the investigator in the same situation as the victim. As a
result, it is often necessary to test a similar surface in an
area not exposed to the dangers of the accident scene. This test
area may not always be on the same grade as the accident scene
and the reconstructionist must adjust for the difference.

The major schools of accident reconstruction generally gloss over the explanation of how to adjust the drag factor for grade, teaching that the acceptable method is to add the signed value of the difference in the percent of grade to the coefficient of kinetic friction (often referred to as drag factor). When the correct method is discussed, it is generally done so in a cursory manner and soon forgotten by the student. This concept pales in comparison to the shear volume of information presented in most courses on accident reconstruction since most of the road surfaces the reconstructionist is likely to encounter have relatively shallow slopes of less than fifteen percent (.15) and the popular method of adjustment will usually suffice. If the slope is greater than ten percent (.1) the error starts to become unacceptable and an alternate method should be used. The strength of the popular method is its simplicity but credence must also be given to those who believe that even though it may be impossible to obtain 100 percent accuracy in every aspect of the reconstruction, every reasonable effort should be made to maintain accuracy. Regardless of the method used, it is beneficial to understand how and why the adjustment should be accomplished.

Gravity effects the coefficient of kinetic friction when it is adjusted for grade in two different ways. The first is in a direction perpendicular to the road surface at the changed grade. This effect can be determined by multiplying by the cosine of the angle of grade (or the cosine of the arctangent of the percent of grade - cos[atn[m]]). The second is in a direction parallel to the surface of the roadway at the changed grade. This effect can be determined by multiplying by the sine of the angle (sine of the arctangent of the percent of grade - sin[atn[m]]). For a positive change of grade this force will act in the same direction as the friction force and therefore must be added to the friction force. The following derivations may assist in understanding this principle

**DERIVATIONS**

E F - sum of the forces acting on an object

E Fx - the sum of the forces on the X axis

E Fy - the sum of the forces on the Y axis

f - coefficient of kinetic friction over a level surface

F - drag factor - the coefficient of kinetic friction (f) adjusted for [braking and/or] grade (m). A negative acceleration factor

fr - friction force

g - the rate of acceleration due to gravity

G - a pure number obtained by dividing the force of gravity by 32.2 ft/sec^2

N - normal force

W - weight

- E Fx = fr + m g sin O = m ax
- E Fy = +N - m g cos O = 0
- fr = f N
- E Fy = f N - f m g cos O = 0 …………………………………………….[multiply eq. 2 by f]
- E Fy = fr - f m g cos O = 0 ………………………………………………[substitute eq. 3 into eq. 4]
- E Fy = f m g cos O - fr = 0 ………………………………………………[multiply eq. 5 by -1]
- E F = f m g cos O + m g sin O = m ax …………………………………..[add eq. 1 to eq. 5]
- E F = m ax = m g (f cos O + sin O) ……………………………………...[re-arrange terms]
- ax = g(f cos O + sin O) …………………………………………………..[divide eq. 8 by m]
- ax = g(f cos O + sin O) …………………………………………………..[acceleration on the X axis]
- F = a/g …………………….……….....[acceleration factor (F) equals acceleration (a) divided by acceleration rate of gravity (g)]
- O = atn m ………………………….....[angle = arctangent of the % of grade (m)]
- F = f cos O + sin O ……………….....[divide eq. 10 by g]
- F = sin(atn m) + f cos(atn m) ……....[substitute atn m for O in eq. 13]

The adjusted drag factor (F) for a positive change of grade, is a combination of both the coefficient of kinetic friction (f) of the object and of the force of gravity or "G" force acting on the object. The coefficient of kinetic friction (f) modified by a positive grade has the force of gravity assisting in the acceleration of the object. Both of these forces combined result in what the reconstructionist refers to as the adjusted drag factor (F).

The following table shows the actual adjusted drag factor (F ACTUAL) along with the drag factor produced with the (f + m) method of approximation. As can be readily observed, the error becomes significant for grades in excess of ten percent (.1). The table also demonstrates that the "drag factor" is not the same for both a positive and negative grade of the same slope.

**COEFFICIENT OF KINETIC FRICTION ADJUSTED FOR
GRADE**

% GRADE ---------------- |
ANGLE --------------- |
f + m---------------------- |
F-ACTUAL --------------- |
% ERROR |

VERTICAL | 90.0000 | INFINITY | 1.0000 | INFINITY |

10000 | 89.4271 | 101.0000 | 1.0099 | 9,900.5000 |

1000 | 84.2894 | 11.0000 | 1.0945 | 904.9876 |

500 | 78.6901 | 6.0000 | 1.1767 | 409.9020 |

200 | 63.4349 | 3.0000 | 1.3416 | 123.6068 |

100 | 45.0000 | 2.0000 | 1.4142 | 41.4241 |

90 | 41.9872 | 1.9000 | 1.4123 | 34.5362 |

80 | 38.6598 | 1.8000 | 1.4056 | 28.0625 |

70 | 34.9920 | 1.7000 | 1.3927 | 22.0656 |

60 | 30.9638 | 1.6000 | 1.3720 | 16.6190 |

50 | 26.5651 | 1.5000 | 1.3416 | 11.8034 |

40 | 21.8014 | 1.4000 | 1.2999 | 7.7033 |

30 | 16.6992 | 1.3000 | 1.2452 | 4.4031 |

20 | 11.3099 | 1.2000 | 1.1767 | 1.9804 |

10 | 5.7106 | 1.1000 | 1.0945 | 0.4988 |

LEVEL | 0.0000 | 1.0000 | 1.0000 | 0.0000 |

-10 | -5.7106 | 0.9000 | 0.8955 | 0.4988 |

-20 | -11.3099 | 0.8000 | 0.7845 | 1.9804 |

-30 | -16.6992 | 0.7000 | 0.6705 | 4.4031 |

-40 | -21.8014 | 0.6000 | 0.5571 | 7.7033 |

-50 | -26.5651 | 0.5000 | 0.4472 | 11.8034 |

-60 | -30.9638 | 0.4000 | 0.3430 | 16.6190 |

-70 | -34.9920 | 0.3000 | 0.2458 | 22.0656 |

-80 | -38.6598 | 0.2000 | 0.1562 | 28.0625 |

-90 | -41.9872 | 0.1000 | 0.0743 | 34.5362 |

-100 | -45.000 | 0.0000 | 0.0000 | 0.0000 |

-200 | -63.4349 | -1.0000 | -0.4472 | 123.6068 |

-500 | -78.6901 | -4.0000 | -0.7845 | 409.9020 |

-1000 | -84.2894 | -9.000 | -0.8955 | 904.9876 |

-10000 | -89.4271 | -99.0000 | -0.9900 | 9,900.5000 |

-INFINITY | -90.000 | -INFINITY | -1.0000 | INFINITY |

The 90 degree vertical position in the above table shows an actual F of 1.0000. This is the force of gravity or "G" force that is accelerating (decelerating) the object as it travels in a positive vertical direction. When the vehicle is level the coefficient of kinetic friction is 1.0000 and therefore the "drag factor" is 1.0000 by either method of computation. The force of gravity is perpendicular to the road surface. The -90 degree vertical position in the table shows an actual F of -1.0000. The surface of the roadway has no effect on the vehicle and gravity is the only force acting upon it. The vehicle is in free fall as no force is acting in opposition to gravity in order to decelerate the vehicle. Since the acceleration factor would be 1.0000 for this condition, the deceleration or negative acceleration factor is -1.0000.

In the determination of the forces acting on a vehicle, the reconstructionist must take into consideration the acceleration mentioned above. This force acts to accelerate the vehicle at any negative angle just as it acts to accelerate (decelerate) the vehicle at any positive angle. The magnitude of this force is obtained by multiplying the acceleration factor by the sine of the negative angle and by the acceleration rate due to gravity of -32.2 ft/sec^2. Therefore, given a level coefficient of friction of 1.0000, a vehicle on a downhill slope of 45 degrees would have an adjusted acceleration or drag factor of zero (0.0000) and a vehicle going downhill vertically would have an acceleration factor of 1.0000 or a drag factor of -1.0000.

It is sometimes necessary to obtain the coefficient of friction for a level surface (f) when the test drag factor (F) is known for the grade. The derivations for the equations used to adjust the test drag factor (F) to level grade are shown below.

15. F - sin O = f cos O ………………………………….[subtract sin O from eq. 13]

16. f = (F - sin O) / cos O ………………………………[divide by cos O and re-arrange terms]

17. f = (F - sin(atn m))/cos(atn m) ……………………..[substitute atn m for O in eq. 16]

If the test drag factor (F) is obtained for a surface that is not level, it must first be adjusted to level. It is only after this adjustment to level has been made that the drag factor on grade can be calculated. It would be mathematically incorrect to adjust directly from one non-level grade to another non-level grade as can be seen in the examples below.

**EXAMPLES**

Example A illustrates the CORRECT method of going from a +.03 test surface with a 1.0000 drag factor to a -.02 grade accident scene:

**Ex. A** f (level) = (F - sin (atn
m))/cos(atn m)

f (level) = (1 - sin (atn .03))/cos(atn .03)

f (level) = .9704

F (@ -.02) = sin (atn m) + f cos (atn m)

F = sin (atn -.02) + .9704 * cos (atn -.02)

F adjusted = .9503

Examples B1 and B2 illustrate two INCORRECT methods of adjusting the drag factor directly for the change of grade (without solving for level), given the identical situation as described above.

**Ex. B1** f = (1 - sin (atn (-.05))/cos(atn
-.05)

f = 1.0512

**Ex. B2** F = sin (atn -.05) + 1 * cos (atn
-.05)

F = .9488

Example C illustrates the "f + m"
method and the resultant INCORRECT solution to the grade problem
in Example A.

**Ex. C** f = 1 + (-.05)

f = .9500

Example C gives a solution very close to the correct solution obtained in example A. This is a result of the minimal grade used in the examples and an "offsetting error condition" in going from a positive test grade to a negative accident grade.

The equations listed are direction sensitive and must be used with the signed (+/-) value of the angle. The positive grade or angle must be used when dealing with a positive slope, regardless of whether the adjustment is from level to the positive slope (eq. 13 or 14) or from the positive slope to level (eq. 16 or 17). When going from level to a negative slope (eq. 13 or 14) or going from a negative slope to level (eq. 16 or 17) a negatively signed angle must be used. The signing of the angle determines if the force of gravity is assisting or resisting in the deceleration. While this last statement may argue the use of a negative angle if going from a positive slope to level, a distinction must be made between a negative slope (going downhill), and returning to level from a positive slope. It would be mathematically incorrect to use a negative angle when returning to level from a positive slope, or to use a positive angle when returning to level from a negative slope.

In this discussion of the forces acting on a vehicle, air resistance has not been mentioned. While air resistance is always acting on a vehicle, because of the relative short distances and the minimal effect this force has on a vehicle traveling at moderate speeds, and the impossibility of determining the velocity of the wind at the time and place of the actual accident, it's effect has not been included in the computations.

The following computer generated results were obtained while tracking two vehicles simultaneously using the f + m method for unit #1 and the F adjusted method for unit #2. The program (REC-TEC, REC-TEC LLC) uses the method outlined above for computation of the adjusted drag factor but can be used to simulate the f + m method. Both vehicles were tracked in a deceleration from an initial velocity of 100 feet per second to a final velocity of zero (0). The base coefficient of friction was one (1.0000) for both vehicles and a negative grade of twenty percent (-.2) was used in the computations.

**TIME/DISTANCE 2**

UNIT #1: DECELERATION ------------------------------------------ | UNIT #2: DECELERATION |

f (TEST) = 1 | f (TEST) = 1 |

GRADE(T) = 0 | GRADE(T) = 0 |

f(LEVEL) = 1 | f(LEVEL) = 1 |

GRADE = -.2 | GRADE = -.2 |

BRAKING = 100 | BRAKING = 100 |

D/FACTOR = .8 | D/FACTOR = .7844 |

RATE OF DECELERATION | RATE OF DECELERATION |

RATE = 25.76 F/S/S | RATE = 25.2597 F/S/S |

RATE = 17.5636 M/H/S | RATE = 17.2225 M/H/S |

DATA (V1 -> V2) | DATA (V1 -> V2) |

DISTANCE = 194.0993 FT | DISTANCE = 197.9433 FT |

TIME = 3.8819 S | TIME = 3.9588 S |

DATA (V1 -> ZERO) | DATA (V1 -> ZERO) |

DISTANCE = 194.0993 FT | DISTANCE = 197.9433 FT |

TIME = 3.8819 S | TIME = 3.9588 S |

INITIAL | INITIAL |

VELOCITY = 100 F/S | VELOCITY = 100 F/S |

VELOCITY = 68.1818 M/H | VELOCITY = 68.1818 M/H |

FINAL | FINAL |

VELOCITY = 0 F/S | VELOCITY = 0 F/S |

VELOCITY = 0 M/H | VELOCITY = 0 M/H |

DISTANCE (FT) | INTERVAL | UNIT #1: 50.0000 | ||||

UNIT #1 | UNIT #2 | |||||

TIME(S) .................... | DIST(FT) .......... | VEL(F/S) ................... | VEL(M/H) ..... | DIST(FT) ..... | VEL(F/S) ..... | VEL(M/H) |

0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

1.9703 | 50.0000 | 50.7543 | 34.6052 | 49.0290 | 49.7687 | 33.9332 |

2.7864 | 100.0000 | 71.7774 | 48.9392 | 98.0581 | 70.3836 | 47.9888 |

3.4126 | 150.0000 | 87.9090 | 59.9380 | 147.0871 | 86.2019 | 58.7740 |

3.8820 | 194.0994 | 100.0000 | 68.1818 | 190.3301 | 98.0581 | 66.8578 |

DISTANCE (FT) | INTERVAL | UNIT #2: 50.0000 | ||||

UNIT #1 | UNIT #2 | |||||

TIME(S) | DIST(FT) | VEL(F/S) | VEL(M/H) | DIST(FT) | VEL(F/S) | VEL(M/H) |

0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

1.9897 | 50.9902 | 51.2544 | 34.9462 | 50.0000 | 50.2591 | 34.2676 |

2.8138 | 101.9804 | 72.4847 | 49.4214 | 100.0000 | 71.0771 | 48.4617 |

3.4462 | 152.9706 | 88.7752 | 60.5286 | 150.0000 | 87.0513 | 59.3532 |

3.9589 | 201.7872 | 100.0000 | 68.1818 | 197.9433 | 100.0000 | 68.1818 |

TIME(S) | INTERVAL | 0.5000 | ||||

UNIT #1 | UNIT #2 | |||||

TIME(S) | DIST(FT) | VEL(F/S) | VEL(M/H) | DIST(FT) | VEL(F/S) | VEL(M/H) |

0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

0.5000 | 3.2200 | 12.8800 | 8.7818 | 3.1575 | 12.6299 | 8.6113 |

1.0000 | 12.8800 | 25.7600 | 17.5636 | 12.6299 | 25.6299 | 17.2226 |

1.5000 | 28.9800 | 38.6400 | 26.3455 | 28.4172 | 37.8896 | 25.8338 |

2.0000 | 51.5200 | 51.5200 | 35.1273 | 50.5195 | 50.5195 | 34.4451 |

2.5000 | 80.5000 | 64.4000 | 43.9091 | 78.9367 | 63.1494 | 43.0564 |

3.0000 | 115.9200 | 77.2800 | 52.6909 | 113.6689 | 75.7793 | 51.6677 |

3.5000 | 157.7800 | 90.1600 | 61.4727 | 154.7160 | 88.4092 | 60.2790 |

3.8820 | 194.0994 | 100.0000 | 68.1818 | 190.3301 | 98.0581 | 66.8578 |

3.9589 | 201.7872 | 100.0000 | 68.1818 | 197.9433 | 100.0000 | 68.1818 |

**CONCLUSIONS**

The equations presented herein can be utilized for adjusting the coefficient of kinetic friction for grade as applied in accident reconstruction. They present an accurate alternative to the approximation method currently in widespread use. The equations and their derivations are a useful tool for those who use either method.

**REFERENCES**

George B. Arfken, David F. Griffing, Donald C.
Kelley, Joseph Priest , **UNIVERSITY PHYSICS
**Miami University, Oxford, Ohio, Academia Press, Inc.

Orlando, Florida, 1984

J. Stanndard Baker, **TRAFFIC ACCIDENT
INVESTIGATION MANUAL
**Northwestern University Traffic Institute

Evanston, Illinois, 1975

John Daily,** FUNDAMENTALS OF TRAFFIC ACCIDENT
RECONSTRUCTION**

Institute of Police Technology and Management

Jacksonville, Florida, 1988

Edward R. McCliment,** PHYSICS**

University of Iowa, Harcourt,Brace Jovanovich, Inc.

Orlando, Florida, 1984

Ross Mocklin, John Rigol, Genovieve May **ADVANCED
TECHNICAL ACCIDENT INVESTIGATION
**Louisiana Department of Public Safety - Office of State
Police

Baton Rouge, Louisiana, 1979

Staff of Research and Education Association, **THE
PHYSICS PROBLEM SOLVER**

Research and Education Association

New York, New York, 1983

**Voice or Fax 1-321-639-7783**

**REC-TEC LLC P.O. BOX 561031
ROCKLEDGE, FL 32956 USA**

**Copyright © George
M. Bonnett, JD**

**Last edited on Monday, 17 January 2022 08:39:45 PM -0500
**