View source for Braking distance

{{Short description|Physics concept relating to automobiles}}
{{Use American English|date = April 2019}}
{{More citations needed|date=January 2007}}

[[File:Vehicle Stopping Distance.webp|thumb|290px|{{center|'''Vehicle Stopping Distance'''}}
{{legend|#B9B9B9|[[Mental chronometry|Reaction Time]] Distance - (3/4) second|outline=#929292}}
{{legend|#FEE47D|[[Passenger vehicle]] Stopping Distance|outline=#FEAE00}}
{{legend|#F57E72|[[List of truck types#Heavy trucks|Heavy]] Truck Stopping Distance|outline=#B51700}}
]]

'''Braking distance''' refers to the distance a vehicle will travel from the point when its [[brakes]] are fully applied to when it comes to a complete stop. It is primarily affected by the original speed of the vehicle and the [[coefficient of friction]] between the [[tire]]s and the [[road surface]],{{#tag:ref |The average friction coefficient (µ) is related to the tire's [[Treadwear rating]] by the following formula:  <math>\mu =\frac{2.25}{TW^{0.15}}</math>   See [http://hpwizard.com/tire-friction-coefficient.htmlg HPwizard on Tire Friction]|group="Note" |name="Treadwear Rating"}} and negligibly by the tires' [[rolling resistance]] and vehicle's [[Automobile drag coefficient|air 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.<ref>{{cite journal |last1=Fricke |first1=L. |title=Traffic Accident Reconstruction: Volume 2 of the Traffic Accident Investigation Manual |publisher=The Traffic Institute, Northwestern University |year=1990}}</ref>{{#tag:ref |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 to [[brake fade]] after extended use, or recover more quickly after water immersion, all should be capable of wheel lock.|group="Note" |name="Effect of brake and vehicle type"}}

The braking distance is one of two principal components of the [[total 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. A [[Mental chronometry|perception-reaction time]] of 1.5 seconds,<ref name="ite.org">{{cite journal |last1=Taoka |first1=George T. |date=March 1989 |title=Brake Reaction Times of Unalerted Drivers |url=https://www.safetylit.org/citations/index.php?fuseaction=citations.viewdetails&citationIds[]=citjournalarticle_241141_38 |journal=ITE Journal |volume=59 |issue=3 |pages=19–21 |issn=0162-8178}}</ref><ref>The [[National Highway Traffic Safety Administration]] (NHTSA) uses 1.5 seconds for the average reaction time.</ref><ref>The [https://archive.today/20130731201913/http://www.vcu.edu/cppweb/tstc/crashinvestigation/ Virginia Commonwealth University’s Crash Investigation Team] typically uses 1.5 seconds to calculate perception-reaction time</ref> and a [[friction#dry friction#kinetic friction|coefficient of kinetic friction]] of 0.7 are standard for the purpose of determining a bare baseline for [[accident reconstruction]] and [[judicial notice]];<ref name="VA Table">{{cite web |url=http://law.lis.virginia.gov/vacode/46.2-880/ |title=Tables of speed and stopping distances |publisher=The State of Virginia}}</ref> most people can stop slightly sooner under ideal conditions.

Braking distance is not to be confused with                      [[stopping sight distance]]. The latter is a [[Geometric design of roads|road alignment]] visibility standard that provides motorists driving at or below the [[design speed]] an [[Assured Clear Distance Ahead|assured clear distance ahead (ACDA)]]<ref>[http://definitions.uslegal.com/a/assured-clear-distance-ahead/ ACDA] or "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</ref> which exceeds a [[safety factor]] distance that would be required by a slightly or [[ordinary care|nearly negligent driver]] to stop under a worst likely case scenario: typically [[Road slipperiness|slippery conditions]] ([[deceleration]] 0.35[[gravity of Earth|g]]<ref>{{cite book |title=NCHRP Report 400: Determination of Stopping Sight Distances |author=National Cooperative Highway Research Program |publisher=Transportation Research Board (National Academy Press) |page=I-13 |year=1997 |isbn=0-309-06073-7 |url=http://onlinepubs.trb.org/onlinepubs/nchrp/nchrp_rpt_400.pdf}}</ref>{{#tag:ref | [https://www.fhwa.dot.gov/programadmin/y2kgb.cfm THE 2001 GREEN BOOK] revised braking distance portion of equation now based on deceleration ( a ) rather than friction factor ( f ) upon recommendation of NCHRP Report 400|group="Note" |name="NCHRP change"}}) and a slow responding driver (2.5 seconds).<ref name="AASHTO">American Association of State Highway and Transportation Officials (1994) ''A Policy on Geometric Design of Highways and Streets'' (Chapter 3)</ref><ref>{{Cite book |url=http://www.dot.ca.gov/hq/oppd/hdm/hdmtoc.htm |title=Highway Design Manual |publisher=California Department of Transportation |volume=6th Ed. |year=2012 |page=200}}  See  [http://www.dot.ca.gov/hq/oppd/hdm/pdf/english/chp0200.pdf Chapter 200 on Stopping Sight Distance] and [http://www.dot.ca.gov/hq/oppd/hdm/pdf/english/chp0400.pdf Chapter 405.1 on Sight Distance]</ref> 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 be [[negligent]] for not stopping sooner.

==Derivation==

===Energy equation===
The theoretical braking distance can be found by determining the [[work (physics)|work]] required to dissipate the vehicle's [[kinetic energy]].<ref>Traffic Accident Reconstruction Volume 2, Lynn B. Fricke</ref>

The kinetic energy {{math|''E''}} is given by the formula:
:<math>E=\frac{1}{2}mv^{2}</math>,
where {{math|''m''}} is the vehicle's mass and {{math|''v''}} is the speed at the start of braking.

The work {{math|''W''}} done by braking is given by:
:<math>W=\mu mgd</math>,
where {{math|μ}} is the [[coefficient of friction]] between the road surface and the tires, {{math|''g''}} is the [[Standard gravity|gravity of Earth]], and {{math|''d''}} is the distance travelled.

The braking distance (which is commonly measured as the skid length) given an initial driving speed {{math|''v''}} is then found by putting {{math|1=''W'' = ''E''}}, from which it follows that
:<math>d=\frac{v^{2}}{2\mu g}</math>.
The maximum speed given an available braking distance {{math|''d''}} is given by:
:<math>v=\sqrt{2\mu gd}</math>.

==Newton's law and equation of motion==
From [[Newton's second law]]:
:<math>F=ma</math>

For a level surface, the [[frictional force#Normal force|frictional force]] resulting from [[coefficient of friction]] <math>\mu</math> is:

:<math>F_{frict}=-\mu mg</math>

Equating the two yields the [[deceleration]]:
:<math>a=-\mu g</math>

The <math>d_f(d_i,v_i,v_f)</math> form of the [[formulas for constant acceleration]] is:
:<math>d_f=d_i + \frac{v_f^2-v_i^2}{2a}</math>

Setting <math>d_i, v_f =0</math> and then [[Change of variables|substituting]] <math>a</math> into the equation yields the braking distance:

:<math>d_f=\frac{-v_i^2}{2a}=\frac{v_i^2}{2 \mu g}</math>

==Total stopping distance==
{{Image frame |caption={{center|''Tables of speed and stopping distances''<ref name="VA Table"/><br>''Permitted by good tires and clean, dry, level, pavement.''}}|content=
{{Graph:Chart |xAxisTitle=Vehicle Speed [mph] |yAxisTitle=Total Stopping Distance [feet] |colors=orange,red |legend=Vehicle |y1Title=Cars |y2Title=Trucks|type=line
|x=10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100
|y1=27,44,63,85,109,135,164,195,229,265,303,344,387,433,481,531,584,639,696
|y2=28,47,69,95,123,155,190,228,269,313,361,411,465,522,582,646,712,782,855
<!--OCTAVE CODE
v=22/15.*[10:5:100];,t=1.5;,mu=0.525;,g=32.255;,d=round(v.*t+v.^2./(2*mu*g))' (mu=0.7 cars; 0.525 trucks) (g=G*1.0025 deduce standard gravity (32.174) or mu appear off by 0.25% in reference)
Run code in online web app here:
https://octave-online.net/
Chart template: https://en.wikipedia.org/wiki/Template:Graph:Chart
Justifications: http://www.virginiadot.org/vtrc/main/online%5Freports/pdf/01-r13.pdf
-->
}}
}}

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

:<math>D_{total}=D_{p-r}+D_{braking}=v t_{p-r}+ \frac{v^2}{2 \mu g}</math>

A common baseline value of <math>t_{p-r}=1.5 s, \mu=0.7</math> is used in stopping distance charts. These values incorporate the ability of the vast majority of drivers under normal road conditions.<ref name="ite.org"/> However, a keen and alert driver may have perception-reaction times well below 1 second,<ref>{{cite web|url=http://biae.clemson.edu/bpc/bp/Lab/110/reaction.htm#Mean%20Times |title=A Literature Review on Reaction Time |author=Robert J. Kosinski |publisher=Clemson University |date=September 2012 |url-status=dead |archive-url=https://web.archive.org/web/20131010194942/http://biae.clemson.edu/bpc/bp/Lab/110/reaction.htm |archive-date=2013-10-10 }}</ref> and a modern car with [[Electronic brakeforce distribution|computerized]] [[Anti-lock braking system|anti-skid brakes]] may have a [[friction#Dry friction|friction coefficient]] of 0.9--or even far exceed 1.0 with sticky tires.<ref name="virginiadot.org">[http://www.virginiadot.org/vtrc/main/online%5Freports/pdf/01-r13.pdf An investigation of the utility and accuracy of the table of speed and stopping distances]  {{webarchive|url=https://web.archive.org/web/20120927004530/http://www.virginiadot.org/vtrc/main/online_reports/pdf/01-r13.pdf |date=September 27, 2012  }}</ref><ref>[http://hpwizard.com/tire-friction-coefficient.html Tire friction and rolling resistance coefficients]</ref><ref>[http://hpwizard.com/gg-diagram.html THE GG DIAGRAM]:  sticky tires exceed 1.0</ref><ref name="J.Y. Wong 1993 26">{{cite book |title=Theory of ground vehicles |volume=2nd ed. |author=J.Y. Wong |year=1993 |page=26 |publisher=John Wiley & Sons |isbn=9780470170380 |url=https://books.google.com/books?id=Blp2D1DteTYC&q=theory+of+ground+vehicles}}</ref><ref>{{cite book |author=Robert Bosch GmbH |title=Automotive Handbook |volume=4th ed |page=335 |year=1996 |publisher=Bentley Publishers |isbn=9780837603339 |url=https://books.google.com/books?id=zsRHPwAACAAJ&q=automotive%20handbook%20bosch}}</ref>

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 a [[two-second rule]] to simulate the elderly or neophyte;{{#tag:ref| A study conducted by the [[Transportation Research Board]] in 1998 found that most people can perceive and react to an unexpected roadway condition in 2 seconds or less. |group="Note" |name="TRB"}} or even a 2.5 second reaction time—to specifically accommodate very elderly, debilitated, intoxicated, or distracted drivers.<ref name="virginiadot.org"/> The coefficient of friction may be 0.25 or lower on wet or frozen asphalt, and [[Anti-lock braking system|anti-skid brakes]] and season specific performance tires may somewhat compensate for driver error and conditions.<ref name="J.Y. Wong 1993 26"/><ref>[http://www.engineeringtoolbox.com/friction-coefficients-d_778.html Frictional Coefficients for some Common Materials and Materials Combinations] and [http://www.engineershandbook.com/Tables/frictioncoefficients.htm Reference Tables -- Coefficient of Friction] {{webarchive|url=https://web.archive.org/web/20090308124246/http://engineershandbook.com/Tables/frictioncoefficients.htm |date=2009-03-08 }}</ref>{{#tag:ref| 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): <math>D_{p-r}=D_{braking}</math> thus <math>vt_{p-r}=\frac{v^2}{2 \mu g}</math>. Solving for v, <math>v=2 \mu g t_{p-r}</math>. This is due to the [[Quadratic function|quadratic]] nature of the kinetic energy increase versus the [[Linear function (calculus)|linear]] effect of a constant p-r time.|group="Note" |name="Speed at which braking distance equals perception-reaction distance"}} In legal contexts, conservative values suggestive of greater minimum stopping distances are often used as to be sure to exceed the pertinent [[legal 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 preponderance [[legal burden of proof#Preponderance of the evidence|more probable than not]], 1.5 seconds is [[legal burden of proof#Clear and convincing|clear and convincing]], and 2.5 seconds is [[legal burden of proof#Beyond reasonable doubt|beyond 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<ref>[http://www.tirerack.com/tires/tests/testSearch.jsp Tire Test Results]</ref> 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 the [[Geometric design of roads|road design]] or [[Reasonable person|expected by other users]], may not be safe to drive.<ref>[http://www.helpguide.org/elder/senior_citizen_driving.htm Warning Signs and Knowing When to Stop Driving] {{webarchive|url=https://web.archive.org/web/20080527023158/http://www.helpguide.org/elder/senior_citizen_driving.htm |date=2008-05-27 }}</ref><ref>{{cite journal |first1=S |last1=Jevas |first2=J. H. |last2=Yan |title=The effect of aging on cognitive function: a preliminary quantitative review. |journal=Research Quarterly for Exercise and Sport |volume=72 |page=A-49 |year=2001}}    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</ref><ref>{{cite journal |first1=G. |last1=Der  |first2=I. J. |last2=Deary |year=2006 |title=Age and sex differences in reaction time in adulthood: Results from the United Kingdom health and lifestyle survey. |journal=Psychology and Aging |volume=21 | issue = 1 |pages=62–73 |doi=10.1037/0882-7974.21.1.62|pmid=16594792 }}</ref> Most old roads were [[Geometric design of roads#Effects of insufficient sight distance|not engineered]] with 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.<ref>{{cite web |url=https://www.fhwa.dot.gov/publications/research/safety/humanfac/01103/ |title=Highway Design Handbook for Older Drivers and Pedestrians |publisher=Publication Number: FHWA-RD-01-103 |date=May 2001}}</ref>

For rubber tyres on cars, the coefficient of friction ({{math|μ}}) decreases as the mass of the car increases. Additionally, {{math|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.<ref>{{cite web|last1=Tomita|first1=Hisao|title=Tire-pavement friction coefficients|url=https://www.dtic.mil/dtic/tr/fulltext/u2/705987.pdf|archive-url=https://web.archive.org/web/20150614022345/http://www.dtic.mil/dtic/tr/fulltext/u2/705987.pdf|url-status=dead|archive-date=June 14, 2015|website=Defense Technical Information Center|publisher=Naval Civil Engineering Laboratory|access-date=12 June 2015}}</ref>

===Rules of thumb===
In a non-[[Metric system|metric]] country, the stopping distance in feet given a velocity in [[miles per hour|MPH]] can 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 = 225 [[foot (unit)|feet]] (the exact value can be calculated using the formula given below the diagram on the right).

In [[Germany]] the 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&nbsp;km/h this corresponds to about 15&nbsp;m. For higher speeds up to about 100&nbsp;km/h outside built-up areas, a similarly defined 2-second rule applies, which for 100&nbsp;km/h translates to about 50&nbsp;m. For speeds on the order of 100&nbsp;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 as ''halber tacho'' (''half the [[speedometer]]'') rule, e.g. for 100&nbsp;km/h the stopping distance should be about 50&nbsp;m. Additionally, German driving schools teach their pupils that the total stopping distance is typically:

<math>(Speed\div10)\times3+(Speed\div10)^2</math>

In the [[UK]], the typical total stopping distances (thinking distance plus braking distance) used in [[The Highway Code]] are quoted in Rule 126 as:<ref>{{Cite web |title=Typical stopping distance |url=https://assets.publishing.service.gov.uk/media/559afb11ed915d1595000017/the-highway-code-typical-stopping-distances.pdf}}</ref>
* 20&nbsp;mph: 40 feet (12 metres)
* 30&nbsp;mph: 75 feet (23 metres)
* 40&nbsp;mph: 118 feet (36 metres)
* 50&nbsp;mph: 175 feet (53 metres)
* 60&nbsp;mph: 240 feet (73 metres)
* 70&nbsp;mph: 315 feet (96 metres)

==See also==
*[[Assured clear distance ahead]]
*[[Brake]]
*[[Cadence braking]]
*[[Following distance]]
*[[Skid mark]]
*[[Stopping sight distance]]
*[[Threshold braking]]
*[[Vehicle metrics]]
*[[Vehicular accident reconstruction]]

==Notes==
{{Reflist|35em|group="Note"}}

==References==
{{Reflist|30em}}

===Further reading===
*{{anchor|84_ALR2d_976}}{{cite book |title=American Law Reports--Annotated, 2nd Series |chapter=Judicial notice of drivers' reaction time and of stopping distance of motor vehicles traveling at various speeds |author=B. Finberg |publisher=The Lawyers Co-operative Publishing Company; Bancroft-Whitney; West Group Annotation Company |volume=84 |page=979 |year=2010}}
*{{anchor|9_ALR3d_976}}{{cite book |title=American Law Reports--Annotated, 3rd Series |chapter=Admissibility in evidence, in automobile negligence action, of charts showing braking distance, reaction times, etc. |author=E. Campion |publisher=The Lawyers Co-operative Publishing Company; Bancroft-Whitney; West Group Annotation Company |volume=9 |page=976 |year=2008}}
*{{anchor|78_ALR2d_218}}{{cite book |title=American Law Reports--Annotated, 2nd Series |chapter=Admissibility of experimental evidence, skidding tests, or the like, relating to speed or control of motor vehicle |author=C. C. Marvel|publisher=The Lawyers Co-operative Publishing Company; Bancroft-Whitney; West Group Annotation Company |volume=78 |page=218 |year=2012}}
*{{anchor|29_ALR3d_248}}{{cite book |title=American Law Reports--Annotated, 3rd Series |chapter=Opinion testimony as to speed of motor vehicle based on skid marks and other facts |author=Jerre E. Box |publisher=The Lawyers Co-operative Publishing Company; Bancroft-Whitney; West Group Annotation Company |volume=29 |page=248 |year=2009}}
*{{anchor|72_ALR2d_6}}{{cite book |title=American Law Reports--Annotated, 2nd Series |chapter=Negligence of driver of motor vehicle as respects manner of timely application of proper brakes |author=Wade R. Habeeb |publisher=The Lawyers Co-operative Publishing Company; Bancroft-Whitney; West Group Annotation Company |volume=72 |page=6 |year=2008}}

==External links==
*[http://www.random-science-tools.com/physics/stopping-distance.htm Car Stopping Distance Calculator]
*[http://forensicdynamics.com/stopping-braking-distance-calculator Braking Distance Calculator]
*[https://web.archive.org/web/20021113014650/http://leg1.state.va.us/cgi-bin/legp504.exe?000+cod+46.2-880 Tables of speed and stopping distances]
*[http://en.wikibooks.org/wiki/Fundamentals_of_Transportation/Sight_Distance Wikibooks: Sight Distance]
*[http://www.gov.uk/highway-code The Highway Code (in English)]

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{{DEFAULTSORT:Braking Distance}}
[[Category:Road transport]]
[[Category:Vehicle braking technologies]]