Braking distance

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 tires and the road surface,^{[Note 1]} and negligibly by the tires' rolling resistance and vehicle's 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.^[1]^{[Note 2]}

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 perception-reaction time of 1.5 seconds,^[2]^[3]^[4] and a coefficient of kinetic friction of 0.7 are standard for the purpose of determining a bare baseline for accident reconstruction and judicial notice;^[5] most people can stop slightly sooner under ideal conditions.

Braking distance is not to be confused with stopping sight distance. The latter is a road alignment visibility standard that provides motorists driving at or below the design speed an assured clear distance ahead (ACDA)^[6] which exceeds a safety factor distance that would be required by a slightly or nearly negligent driver to stop under a worst likely case scenario: typically slippery conditions (deceleration 0.35g^[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 be negligent for not stopping sooner.

Derivation

Energy equation

The theoretical braking distance can be found by determining the work required to dissipate the vehicle's kinetic 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 the coefficient of friction between the road surface and the tires, $g$ is the gravity 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

From Newton's second law:

F=ma

For a level surface, the frictional force resulting from coefficient of friction $\mu$ is:

F_{frict}=-\mu mg

Equating the two yields the deceleration:

a=-\mu g

The $d_{f}(d_{i},v_{i},v_{f})$ form of the formulas for constant acceleration is:

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

Setting $d_{i},v_{f}=0$ and then substituting $a$ 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 with computerized anti-skid brakes may have a friction coefficient of 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 a two-second rule to 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, and anti-skid brakes and 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 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 more probable than not, 1.5 seconds is clear and convincing, and 2.5 seconds is 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^[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 the road design or expected by other users, may not be safe to drive.^[19]^[20]^[21] Most old roads were 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.^[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-metric country, the stopping distance in feet given a velocity in 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 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 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 as halber tacho (half the speedometer) 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 the UK, the typical total stopping distances (thinking distance plus braking distance) used in The Highway Code are 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's Treadwear rating by the following formula: $\mu ={\frac {2.25}{TW^{0.15}}}$ See HPwizard 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 to brake fade after extended use, or recover more quickly after water immersion, all should be capable of wheel lock.
^ 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
^ 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.
^ 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 the quadratic nature of the kinetic energy increase versus the linear effect 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 ^b Taoka, George T. (March 1989). "Brake Reaction Times of Unalerted Drivers". ITE Journal. 59 (3): 19–21. ISSN 0162-8178.
^ The National Highway Traffic Safety Administration (NHTSA) uses 1.5 seconds for the average reaction time.
^ The Virginia Commonwealth University’s Crash Investigation Team typically uses 1.5 seconds to calculate perception-reaction time
^ ^a ^b "Tables of speed and stopping distances". The State of Virginia.
^ 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
^ 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. See Chapter 200 on Stopping Sight Distance and Chapter 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 from the original on 2013-10-10.
^ ^a ^b An investigation of the utility and accuracy of the table of speed and stopping distances Archived September 27, 2012, at the Wayback Machine
^ Tire friction and rolling resistance coefficients
^ THE GG DIAGRAM: sticky tires exceed 1.0
^ ^a ^b J.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 Combinations and Reference Tables -- Coefficient of Friction Archived 2009-03-08 at the Wayback Machine
^ Tire Test Results
^ Warning Signs and Knowing When to Stop Driving Archived 2008-05-27 at the Wayback 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 from the original (PDF) on June 14, 2015. Retrieved 12 June 2015.
^ "Typical stopping distance" (PDF).

External links

[Treadwear_Rating-1] The average friction coefficient (µ) is related to the tire's Treadwear rating by the following formula: $\mu ={\frac {2.25}{TW^{0.15}}}$ See HPwizard 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 to brake fade after extended use, or recover more quickly after water immersion, all should be capable of wheel lock.

[NCHRP_change-10] 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

[TRB-20] 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.

[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 the quadratic nature of the kinetic energy increase versus the linear effect 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] The National Highway Traffic Safety Administration (NHTSA) uses 1.5 seconds for the average reaction time.

[6] The Virginia Commonwealth University’s Crash Investigation Team typically uses 1.5 seconds to calculate perception-reaction time

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

[8] 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

[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. See Chapter 200 on Stopping Sight Distance and Chapter 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 from the original on 2013-10-10.

[virginiadot.org-15] An investigation of the utility and accuracy of the table of speed and stopping distances Archived September 27, 2012, at the Wayback 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 Combinations and Reference Tables -- Coefficient of Friction Archived 2009-03-08 at the Wayback Machine

[23] Tire Test Results

[24] Warning Signs and Knowing When to Stop Driving Archived 2008-05-27 at the Wayback 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 from the original (PDF) on June 14, 2015. Retrieved 12 June 2015.

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

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