Braking distance

Braking distancerefers to the distance a vehicle will travel from the point when itsbrakesare fully applied to when it comes to a complete stop. It is primarily affected by the original speed of the vehicle and thecoefficient of frictionbetween thetiresand theroad surface,^{[Note 1]}and negligibly by the tires'rolling resistanceand vehicle'sair drag.The type of brake system in use only affects trucks and large mass vehicles, which cannot supply enough force to match the static frictional force.^[1]^{[Note 2]}

The braking distance is one of two principal components of thetotal stopping distance.The other component is the reaction distance, which is the product of the speed and the perception-reaction time of the driver/rider. Aperception-reaction timeof 1.5 seconds,^[2]^[3]^[4]and acoefficient of kinetic frictionof 0.7 are standard for the purpose of determining a bare baseline foraccident reconstructionandjudicial notice;^[5]most people can stop slightly sooner under ideal conditions.

Braking distance is not to be confused withstopping sight distance.The latter is aroad alignmentvisibility standard that provides motorists driving at or below thedesign speedanassured clear distance ahead (ACDA)^[6]which exceeds asafety factordistance that would be required by a slightly ornearly negligent driverto stop under a worst likely case scenario: typicallyslippery conditions(deceleration0.35g^[7]^{[Note 3]}) and a slow responding driver (2.5 seconds).^[8]^[9]Because the stopping sight distance far exceeds the actual stopping distance under most conditions, an otherwise capable driver who uses the full stopping sight distance, which results in injury, may benegligentfor not stopping sooner.

Derivation

Energy equation

The theoretical braking distance can be found by determining theworkrequired to dissipate the vehicle'skinetic energy.^[10]

The kinetic energy $E$ is given by the formula:

E={\frac {1}{2}}mv^{2}

,

where $m$ is the vehicle's mass and $v$ is the speed at the start of braking.

The work $W$ done by braking is given by:

W=\mu mgd

,

where $μ$ is thecoefficient of frictionbetween the road surface and the tires, $g$ is thegravity of Earth,and $d$ is the distance travelled.

The braking distance (which is commonly measured as the skid length) given an initial driving speed $v$ is then found by putting $W = E$ ,from which it follows that

d={\frac {v^{2}}{2\mu g}}

.

The maximum speed given an available braking distance $d$ is given by:

v={\sqrt {2\mu gd}}

.

Newton's law and equation of motion

FromNewton's second law:

F=ma

For a level surface, thefrictional forceresulting fromcoefficient of friction $\mu$ is:

F_{frict}=-\mu mg

Equating the two yields thedeceleration:

a=-\mu g

The $d_{f}(d_{i},v_{i},v_{f})$ form of theformulas for constant accelerationis:

d_{f}=d_{i}+{\frac {v_{f}^{2}-v_{i}^{2}}{2a}}

Setting $d_{i},v_{f}=0$ and thensubstituting $a$ into the equation yields the braking distance:

d_{f}={\frac {-v_{i}^{2}}{2a}}={\frac {v_{i}^{2}}{2\mu g}}

Total stopping distance

Tables of speed and stopping distances^[5]
Permitted by good tires and clean, dry, level, pavement.

The total stopping distance is the sum of the perception-reaction distance and the braking distance.

D_{total}=D_{p-r}+D_{braking}=vt_{p-r}+{\frac {v^{2}}{2\mu g}}

A common baseline value of $t_{p-r}=1.5s,\mu =0.7$ is used in stopping distance charts. These values incorporate the ability of the vast majority of drivers under normal road conditions.^[2]However, a keen and alert driver may have perception-reaction times well below 1 second,^[11]and a modern car withcomputerized anti-skid brakesmay have afriction coefficientof 0.9--or even far exceed 1.0 with sticky tires.^[12]^[13]^[14]^[15]^[16]

Experts historically used a reaction time of 0.75 seconds, but now incorporate perception resulting in an average perception-reaction time of: 1 second for population as an average; occasionally atwo-second ruleto simulate the elderly or neophyte;^{[Note 4]}or even a 2.5 second reaction time—to specifically accommodate very elderly, debilitated, intoxicated, or distracted drivers.^[12]The coefficient of friction may be 0.25 or lower on wet or frozen asphalt, andanti-skid brakesand season specific performance tires may somewhat compensate for driver error and conditions.^[15]^[17]^{[Note 5]}In legal contexts, conservative values suggestive of greater minimum stopping distances are often used as to be sure to exceed the pertinentlegal burden of proof,with care not to go as far as to condone negligence. Thus, the reaction time chosen can be related to the burden's corresponding population percentile; generally a reaction time of 1 second is as a preponderancemore probable than not,1.5 seconds isclear and convincing,and 2.5 seconds isbeyond reasonable doubt.The same principle applies to the friction coefficient values.

Actual total stopping distance

The actual total stopping distance may differ from the baseline value when the road or tire conditions are substantially different from the baseline conditions, or when the driver's cognitive function is superior or deficient. To determine actual total stopping distance, one would typically empirically obtain the coefficient of friction between the tire material^[18]and the exact road spot under the same road conditions and temperature. They would also measure the person's perception and reaction times. A driver who has innate reflexes, and thus braking distances, that are far below the safety margins provided in theroad designorexpected by other users,may not be safe to drive.^[19]^[20]^[21]Most old roads werenot engineeredwith the deficient driver in mind, and often used a defunct 3/4 second reaction time standard. There have been recent road standard changes to make modern roadways more accessible to an increasingly aging population of drivers.^[22]

For rubber tyres on cars, the coefficient of friction ( $μ$ ) decreases as the mass of the car increases. Additionally, $μ$ depends on whether the wheels are locked or rolling during the braking, and a few more parameters such as rubber temperature (increases during the braking) and speed.^[23]

Rules of thumb

In a non-metriccountry, the stopping distance in feet given a velocity inMPHcan be approximated as follows:

take the first digit of the velocity, and square it. Add a zero to the result, then divide by 2.
sum the previous result to the double of the velocity.

Example: velocity = 50 MPH. stopping distance = 5 squared = 25, add a zero = 250, divide by 2 = 125, sum 2*50 = 225feet(the exact value can be calculated using the formula given below the diagram on the right).

InGermanythe rule of thumb for the stopping distance in a city in good conditions is the 1-second rule, i.e. the distance covered in 1 second should at most be the distance to the vehicle ahead. At 50 km/h this corresponds to about 15 m. For higher speeds up to about 100 km/h outside built-up areas, a similarly defined 2-second rule applies, which for 100 km/h translates to about 50 m. For speeds on the order of 100 km/h there is also the more or less equivalent rule that the stopping distance be the speed divided by 2 k/h, referred to ashalber tacho(half thespeedometer) rule, e.g. for 100 km/h the stopping distance should be about 50 m. Additionally, German driving schools teach their pupils that the total stopping distance is typically:

$(Speed\div 10)\times 3+(Speed\div 10)^{2}$

In theUK,the typical total stopping distances (thinking distance plus braking distance) used inThe Highway Codeare quoted in Rule 126 as:^[24]

20 mph: 40 feet (12 metres)
30 mph: 75 feet (23 metres)
40 mph: 118 feet (36 metres)
50 mph: 175 feet (53 metres)
60 mph: 240 feet (73 metres)
70 mph: 315 feet (96 metres)

Notes

^The average friction coefficient (µ) is related to the tire'sTreadwear ratingby the following formula: $\mu ={\frac {2.25}{TW^{0.15}}}$ SeeHPwizard on Tire Friction
^The coefficient of friction is the ratio of the force necessary to move one body horizontally over another at a constant speed to the weight of the body. For a 10 ton truck, the force necessary to lock the brakes could be 7 tons, which is enough force to destroy the brake mechanism itself. While some brake types on lightweight vehicles are more prone tobrake fadeafter extended use, or recover more quickly after water immersion, all should be capable of wheel lock.
^THE 2001 GREEN BOOKrevised braking distance portion of equation now based on deceleration ( a ) rather than friction factor ( f ) upon recommendation of NCHRP Report 400
^A study conducted by theTransportation Research Boardin 1998 found that most people can perceive and react to an unexpected roadway condition in 2 seconds or less.
^As speed increases, the braking distance is initially far less than the perception-reaction distance, but later it equals then rapidly exceeds it after 30 MPH for 1 second p-t times (46 MPH for 1.5s p-t times): $D_{p-r}=D_{braking}$ thus $vt_{p-r}={\frac {v^{2}}{2\mu g}}$ .Solving for v, $v=2\mu gt_{p-r}$ .This is due to thequadraticnature of the kinetic energy increase versus thelineareffect of a constant p-r time.

References

^Fricke, L. (1990). "Traffic Accident Reconstruction: Volume 2 of the Traffic Accident Investigation Manual". The Traffic Institute, Northwestern University.{{cite journal}}:Cite journal requires|journal=(help)
^^a ^bTaoka, George T. (March 1989)."Brake Reaction Times of Unalerted Drivers".ITE Journal.59(3): 19–21.ISSN 0162-8178.
^TheNational Highway Traffic Safety Administration(NHTSA) uses 1.5 seconds for the average reaction time.
^TheVirginia Commonwealth University’s Crash Investigation Teamtypically uses 1.5 seconds to calculate perception-reaction time
^^a ^b"Tables of speed and stopping distances".The State of Virginia.
^ACDAor "assured clear distance ahead" rule requires a driver to keep his vehicle under control so that he can stop in the distance in which he can see clearly
^National Cooperative Highway Research Program (1997).NCHRP Report 400: Determination of Stopping Sight Distances(PDF).Transportation Research Board (National Academy Press). p. I-13.ISBN 0-309-06073-7.
^American Association of State Highway and Transportation Officials (1994)A Policy on Geometric Design of Highways and Streets(Chapter 3)
^Highway Design Manual.Vol. 6th Ed. California Department of Transportation. 2012. p. 200.SeeChapter 200 on Stopping Sight DistanceandChapter 405.1 on Sight Distance
^Traffic Accident Reconstruction Volume 2, Lynn B. Fricke
^Robert J. Kosinski (September 2012)."A Literature Review on Reaction Time".Clemson University. Archived fromthe originalon 2013-10-10.
^^a ^bAn investigation of the utility and accuracy of the table of speed and stopping distances ArchivedSeptember 27, 2012, at theWayback Machine
^Tire friction and rolling resistance coefficients
^THE GG DIAGRAM:sticky tires exceed 1.0
^^a ^bJ.Y. Wong (1993).Theory of ground vehicles.Vol. 2nd ed. John Wiley & Sons. p. 26.ISBN 9780470170380.
^Robert Bosch GmbH (1996).Automotive Handbook.Vol. 4th ed. Bentley Publishers. p. 335.ISBN 9780837603339.
^Frictional Coefficients for some Common Materials and Materials CombinationsandReference Tables -- Coefficient of Friction Archived2009-03-08 at theWayback Machine
^Tire Test Results
^Warning Signs and Knowing When to Stop Driving Archived2008-05-27 at theWayback Machine
^Jevas, S; Yan, J. H. (2001). "The effect of aging on cognitive function: a preliminary quantitative review".Research Quarterly for Exercise and Sport.72:A-49.Simple reaction time shortens from infancy into the late 20s, then increases slowly until the 50s and 60s, and then lengthens faster as the person gets into his 70s and beyond
^Der, G.; Deary, I. J. (2006). "Age and sex differences in reaction time in adulthood: Results from the United Kingdom health and lifestyle survey".Psychology and Aging.21(1): 62–73.doi:10.1037/0882-7974.21.1.62.PMID 16594792.
^"Highway Design Handbook for Older Drivers and Pedestrians".Publication Number: FHWA-RD-01-103. May 2001.
^Tomita, Hisao."Tire-pavement friction coefficients"(PDF).Defense Technical Information Center.Naval Civil Engineering Laboratory. Archived fromthe original(PDF)on June 14, 2015.Retrieved12 June2015.
^"Typical stopping distance"(PDF).

External links

[Treadwear_Rating-1] The average friction coefficient (µ) is related to the tire'sTreadwear ratingby the following formula: $\mu ={\frac {2.25}{TW^{0.15}}}$ SeeHPwizard on Tire Friction

[Effect_of_brake_and_vehicle_type-3] The coefficient of friction is the ratio of the force necessary to move one body horizontally over another at a constant speed to the weight of the body. For a 10 ton truck, the force necessary to lock the brakes could be 7 tons, which is enough force to destroy the brake mechanism itself. While some brake types on lightweight vehicles are more prone tobrake fadeafter extended use, or recover more quickly after water immersion, all should be capable of wheel lock.

[NCHRP_change-10] THE 2001 GREEN BOOKrevised braking distance portion of equation now based on deceleration ( a ) rather than friction factor ( f ) upon recommendation of NCHRP Report 400

[TRB-20] A study conducted by theTransportation Research Boardin 1998 found that most people can perceive and react to an unexpected roadway condition in 2 seconds or less.

[Speed_at_which_braking_distance_equals_perception-reaction_distance-22] As speed increases, the braking distance is initially far less than the perception-reaction distance, but later it equals then rapidly exceeds it after 30 MPH for 1 second p-t times (46 MPH for 1.5s p-t times): $D_{p-r}=D_{braking}$ thus $vt_{p-r}={\frac {v^{2}}{2\mu g}}$ .Solving for v, $v=2\mu gt_{p-r}$ .This is due to thequadraticnature of the kinetic energy increase versus thelineareffect of a constant p-r time.

[2] Fricke, L. (1990). "Traffic Accident Reconstruction: Volume 2 of the Traffic Accident Investigation Manual". The Traffic Institute, Northwestern University.{{cite journal}}:Cite journal requires|journal=(help)

[ite.org-4] Taoka, George T. (March 1989)."Brake Reaction Times of Unalerted Drivers".ITE Journal.59(3): 19–21.ISSN 0162-8178.

[5] TheNational Highway Traffic Safety Administration(NHTSA) uses 1.5 seconds for the average reaction time.

[6] TheVirginia Commonwealth University’s Crash Investigation Teamtypically uses 1.5 seconds to calculate perception-reaction time

[VA_Table-7] "Tables of speed and stopping distances".The State of Virginia.

[8] ACDAor "assured clear distance ahead" rule requires a driver to keep his vehicle under control so that he can stop in the distance in which he can see clearly

[9] National Cooperative Highway Research Program (1997).NCHRP Report 400: Determination of Stopping Sight Distances(PDF).Transportation Research Board (National Academy Press). p. I-13.ISBN 0-309-06073-7.

[AASHTO-11] American Association of State Highway and Transportation Officials (1994)A Policy on Geometric Design of Highways and Streets(Chapter 3)

[12] Highway Design Manual.Vol. 6th Ed. California Department of Transportation. 2012. p. 200.SeeChapter 200 on Stopping Sight DistanceandChapter 405.1 on Sight Distance

[13] Traffic Accident Reconstruction Volume 2, Lynn B. Fricke

[14] Robert J. Kosinski (September 2012)."A Literature Review on Reaction Time".Clemson University. Archived fromthe originalon 2013-10-10.

[virginiadot.org-15] An investigation of the utility and accuracy of the table of speed and stopping distances ArchivedSeptember 27, 2012, at theWayback Machine

[16] Tire friction and rolling resistance coefficients

[17] THE GG DIAGRAM:sticky tires exceed 1.0

[J.Y._Wong_1993_26-18] J.Y. Wong (1993).Theory of ground vehicles.Vol. 2nd ed. John Wiley & Sons. p. 26.ISBN 9780470170380.

[19] Robert Bosch GmbH (1996).Automotive Handbook.Vol. 4th ed. Bentley Publishers. p. 335.ISBN 9780837603339.

[21] Frictional Coefficients for some Common Materials and Materials CombinationsandReference Tables -- Coefficient of Friction Archived2009-03-08 at theWayback Machine

[23] Tire Test Results

[24] Warning Signs and Knowing When to Stop Driving Archived2008-05-27 at theWayback Machine

[25] Jevas, S; Yan, J. H. (2001). "The effect of aging on cognitive function: a preliminary quantitative review".Research Quarterly for Exercise and Sport.72:A-49.Simple reaction time shortens from infancy into the late 20s, then increases slowly until the 50s and 60s, and then lengthens faster as the person gets into his 70s and beyond

[26] Der, G.; Deary, I. J. (2006). "Age and sex differences in reaction time in adulthood: Results from the United Kingdom health and lifestyle survey".Psychology and Aging.21(1): 62–73.doi:10.1037/0882-7974.21.1.62.PMID 16594792.

[27] "Highway Design Handbook for Older Drivers and Pedestrians".Publication Number: FHWA-RD-01-103. May 2001.

[28] Tomita, Hisao."Tire-pavement friction coefficients"(PDF).Defense Technical Information Center.Naval Civil Engineering Laboratory. Archived fromthe original(PDF)on June 14, 2015.Retrieved12 June2015.

[29] "Typical stopping distance"(PDF).

[Note 1]

[1]

[Note 2]

[2]

[3]

[4]

[5]

[6]

[7]

[Note 3]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[Note 4]

[17]

[Note 5]

[18]

[19]

[20]

[21]

[22]

[23]

[24]