The Assured Clear Distance Ahead (ACDA) is the distance ahead of a vehicle or craft which can be seen to be clear of hazards by the driver, within which they should be able to bring the vehicle to a halt.^{[1]} It is one of the most fundamental principles governing ordinary care and the duty of care and is frequently used to determine if a driver is in proper control and is a nearly universally implicit consideration in vehicular accident liability.^{[2]}^{[3]}^{[4]}
This distance is typically both determined and constrained by the proximate edge of clear visibility, but it may be attenuated to a margin of which beyond hazards may reasonably be expected to spontaneously appear. It is a spatial component to the common law basic speed rule. The twosecond rule may be the limiting factor governing the ACDA, when the speed of forward traffic is what limits the basic safe speed, and a primary hazard of collision could result from following any closer.
ACDA as common law rule or statute
"At common law a motorist is required to regulate his speed so that he can stop within the range of his vision. In numerous jurisdictions, this rule has been incorporated in statutes which typically require that no person shall drive any motor vehicle in and upon any public road or highway at a greater speed than will permit him to bring it to a stop within the assured clear distance ahead."^{[2]} Decisional law usually settles the circumstances by which a portion of the roadway is assuredly clear without it being mentioned in statute.^{[5]} California is a state where the judiciary has established the state's ACDA law.^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[Note 1]} Most state issued driver handbooks either instruct or mention the ACDA rule as required care or safe practice.^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}
Many states have further passed statutes which require their courts to more inflexibly weigh the ACDA in their determination of reasonable speed or behavior. Such statutes do so in part by designating ACDA violations as a citable driving offense, thus burdening an offending driver to rebut a presumption of negligence. States with such explicit ACDA standard of care provisions include: Iowa,^{[16]} Michigan,^{[17]} Ohio,^{[18]} Oklahoma,^{[19]} and Pennsylvania.^{[20]} States which apply the principle by statute to watercraft on navigable waterways include: Montana,^{[21]} Florida,^{[22]} Louisiana,^{[23]} and West Virginia. Explicit ACDA statutes, especially those of which create a citable driving offense, are aimed at preventing harm that could result from potentially negligent behavior—whereas the slightly more obscure common law ACDA doctrine is most easily invoked to remedy actual damages that have already occurred as a result of such negligence. Explicit and implicit ACDA rules govern millions of North American drivers.
Determining the ACDA
Static ACDA
Forward "lineofsight" distance
The range of visibility of which is the de facto ACDA, is usually that distance before which an ordinary person can see small hazards—such as a traffic cone or buoy—with 20/20 vision. This distance may be attenuated by specific conditions such as atmospheric opacity,^{[9]} blinding glare,^{[24]} darkness,^{[10]} road design,^{[8]}^{[25]} and adjacent environmental hazards including civil and recreational activities,^{[26]} deer, livestock, crossing traffic,^{[27]} and parked cars. The ACDA may also be somewhat attenuated on roads with lower functional classification.^{[25]} ^{[26]} This is because the probability of spontaneous traffic increases proportionally to the density of road access points, and this density reduces the distance a person exercising ordinary care can be assured that a road will be clear; such reduction in the ACDA is readily apparent from the conditions, even when a specific access point or the traffic thereon is not.^{[28]}^{[Note 2]} Furthermore, even though a throughdriver may typically presume all traffic will stay assuredly clear when required by law, such driver may not take such presumption when circumstances provide actual knowledge under ordinary care that such traffic cannot obey the law.^{[28]}
Intersections
Where there are cross roads or side roads with view obstructions, the assured clear distance terminates at the closest path of potential users of the roadway until there is such a view which assures the intersection will remain clear. In such situations, approach speed must be reduced in preparation for entering or crossing a road or intersection or the unmarked pedestrian crosswalks^{[29]}^{[30]} and bike paths^{[31]} they create because of potential hazards.^{[32]}^{[33]}^{[34]}^{[35]}
When approaching an unsignalized intersection controlled by a stop sign, the assured clear distance ahead is:

ACDA_{si}=V ( \sqrt{\frac{2d_i}{a_i}} + t_{pc} )
Normal acceleration "a_{i}" for a passenger vehicle from a stop up to 20 mph is about 0.15g, with more than 0.3g being difficult to exceed.^{[32]} The distance "d_{i}" is the sum of the measured limit line setback distance—which is typically regulated by a Manual on Uniform Traffic Control Devices, at often between 4 and 30 feet in the United States^{[36]}^{[37]}—and the crosswalk, parking lane, and road shoulder width. A vehicle accelerating from a stop travels this distance in time "t_{i}=√(2d_{i}/a_{i})" while through traffic travels a distance equal to their speed multiplied by that time. The time t_{pc}, for the stopped motorist, is the sum of perception time and the time required to actuate an automatic transmission or shift to first gear which is usually between 1/2 to one second.^{[38]}
ACDA as a function of horizontal sight distance
Horizontal clearance is measured from the edge of the traveled way to the bottom of the nearest object, tree trunk or shrub foliage mass face, plant setback, or mature growth.^{[39]}^{[40]} Horizontal sight distance is not to be confused with the clear recovery zone which provides hazardous vegetation setback to allow errant vehicles to regain control, and is exclusive to a mowed and limbedup forest which can allow adequate sight distance, but unsafe recovery.^{[40]} The height and lateral distance of plants restrict the horizontal sight distance, at times obscuring wildlife which may be spooked by an approaching vehicle and run across the road to escape with their herd.^{[40]}^{[41]} This principle also applies to approaching vehicles and pedestrians at uncontrolled intersections and to a lesser degree by unsignalized intersections controlled by a yield sign. Horizontal sight distance "d_{hsd}" affects the ACDA because the time "t_{i}=d_{hsd}/V_{i}" it takes for an intercepting object, animal, pedestrian, or vehicle with speed "V_{i}" to transverse this distance after emerging from the proximate edge of lateral visibility affords a vehicle with speed "V" a clear distance of "Vt_{i}." Thus, the assured clear intercept distance "ACDA_{si}" is:

ACDA_{si}=\frac{V d_{hsd}}{V_i}
The faster one drives, the farther downroad an interceptor must be in order to be able to transverse the horizontal sight distance in time to collide, however this says nothing of whether the vehicle can stop by the end of this type of assured clear distance. Equating this distance to the total stopping distance and solving for speed yields one's maximum safe speed as purely dictated by the horizontal sight distance.
Dynamic "following" distance
The ACDA may also be dynamic as to the moving distance past which a motorist can be assured be to able to stay clear of a foreseeable dynamic hazard—such as to maintain a distance as to be able to safely swerve around a bicyclist should he succumb to a fall—without requiring a full stop beforehand, if doing so could be exercised with due care towards surrounding traffic. Quantitatively this distance is a function of the appropriate time gap and the operating speed: d_{ACDA}=t_{gap}*v. The assured clear distance ahead rule, rather than being subject to exceptions, is not really intended to apply beyond situations in which a vigilant ordinarily prudent person could or should anticipate.^{[2]} A common way to violate the dynamic ACDA is by tailgating.
Measurement
The most accurate way to determine the ACDA is to directly measure it. Whereas this is impractical, sight distance formulas can be used with less direct measurements as rough baseline estimates. The empirical assured clear distance ahead calculated with computer vision, range finding, traction control, and GIS, such as by properly programming computer hardware used in autonomous cars, can be recorded to later produce or color baseline ACDA and safe speed maps for accident investigation, traffic engineering, and show disparities between safe speed and 85th percentile "operating" speed. Selfdriving cars may have a higher safe speed than human driven vehicles for a given ACDA where computer perceptionreaction times are nearly instantaneous.
Discretion
The Assured Clear Distance Ahead can be subjective to the baseline estimate of a reasonable person or be predetermined by law. For example, whether one should have reasonably foreseen that a road was not assuredly clear past 75–100 meters because of tractors or livestock which commonly emerge from encroaching blinding vegetation is on occasion dependent on societal experience within the locale. In certain urban environments, a straight, trafficless, throughstreet may not necessarily be assuredly clear past the entrance of the nearest visually obstructed intersection as law.^{[25]}^{[26]}^{[28]} Within the assured clear distance ahead, there is certainty that travel will be free from obstruction which is exclusive of a failure to appreciate a hazard. Collisions generally only occur within one's assured clear distance ahead which are "unavoidable" to them such that they have zero comparative negligence including legal acts of god and abrupt unforeseeably wanton negligence by another party. Hazards which penetrate one's proximate edge of clear visibility and compromise their ACDA generally require evasive action.
Drivers need not and are not required to precisely determine the maximum safe speed from realtime mathematical calculations of sight distances and stopping distances for their particular vehicle. Motor vehicle operators of average intelligence constantly utilize their kinesthetic memory in all sorts of driving tasks including every time they break to a full stop at a stop line in a panoply of conditions. Like throwing a softball, one does not have to calculate a trajectory or firing solution in order to hit a target with repeated accuracy. During the earliest stages of learning how to drive, one develops a memory of when to start breaking (how long it takes) from various speeds in order to stop at the limit line. While there may be a degree of variance of such skill in drivers, they generally do not have the discretion in engaging in a behavior such as driving a speed above which no reasonable minds might differ as to whether it is unsafe or that one could come to a stop within the full distance ahead.^{[35]}
Relation to the basic speed rule
The ACDA distances are a principal component to be evaluated in the determination of the maximum safe speed (V_{BSL}) under the basic speed law, without which the maximum safe speed cannot be determined. The relation of the ACDA to the basic speed rule for land based vehicles may be objectively quantified as follows:

V_{BSL}= \begin{cases} \sqrt{(\mu +e)^2 g^2 t_{prt}^2+ 2 (\mu + e) g d_{ACDA_s} }  (\mu+e) g t_{prt}, & \mbox{if } V_{ACDA_s} \le V_{ACDA_{si1}} \mbox{ or } V_{ACDA_{si2}} \mbox{ or } V_{ACDA_d} \mbox{ or } V_{cs} \mbox{ or } V_{cl}\\ \\ 2 g (\mu + e) (\frac{d_{hsd}}{v_i}t_{prt}), & \mbox{if } V_{ACDA_{si1}} < V_{ACDA_s} \mbox{ or } V_{ACDA_{si2}} \mbox{ or } V_{ACDA_d} \mbox{ or } V_{cs} \mbox{ or } V_{cl}\\ \\ 2 g (\mu + e )(\sqrt{\frac{2 d_{sl}}{a_i}}+t_{pc}t_{prt}), & \mbox{if } V_{ACDA_{si2}} < V_{ACDA_s} \mbox{ or } V_{ACDA_{si1}} \mbox{ or } V_{ACDA_d} \mbox{ or } V_{cs} \mbox{ or } V_{cl}\\ \\ \frac{d_{ACDA_d}}{t_g}, & \mbox{if } V_{ACDA_d}< V_{ACDA_s} \mbox{ or } V_{ACDA_{si1}} \mbox{ or } V_{ACDA_{si2}} \mbox{ or } V_{cs} \mbox{ or } V_{cl} \\ \\ \sqrt{ \frac{(\mu+e) g r}{1\mu e}}, & \mbox{if } V_{cs}
The value of the variable "e" is the sine of the angle of inclination of the road's slope. For a level road this value is zero, and for small angles it approximates the road's percent grade divided by one hundred.

e=sin(\theta) \approx \theta \approx tan(\theta) = \frac{%grade}{100}
^{[Note 3]}
The maximum velocity permitted by the Assured Clear Distance Ahead is controlling of safe speed (V_{BSL}) for only the top and two cases. Safe speed may be greater or less than the actual legal speed limit depending upon the conditions along the road.^{[32]}
^{[Note 4]}
See reference V_{BSL} derivations for basic physics explanation.
ACDA: forward lineofsight
For the top case, the maximum speed is governed by the assured clear "lineofsight", as when the "following distance" aft of forward traffic and "steering control" are both adequate. Common examples include when there is no vehicle to be viewed, or when there is a haze or fog that would prevent visualizing a close vehicle in front. This maximum velocity is denoted by the case variable V_{ACDA_s}, the friction coefficient is symbolized by \mu—and itself a function of the tire type and road conditions, the distance d_{ACDA_s} is the static ACDA, the constant g is the acceleration of gravity, and interval t_{prt} is the perceptionreaction time—usually between 1.0 and 2.5 seconds.^{[42]}
ACDA: horizontal lineofsight
The second case describes the relationship of horizontal sight distance on safe speed. It is the maximum speed at which a vehicle can come to a full stop before an object, with speed V_{i}, can intercept after having emerged and traveled across the horizontal sight distance "d_{hsd}." Urban and residential areas have horizontal sight distance that tends to be closely obstructed by parked cars, utility poles, street furnishing, fencing, signage, and landscaping, but have slower intercepting speeds of children, pedestrians, backing cars, and domestic animals. These interceptors combined with dense usage results in collisions that are more probable and much more likely to inflict harm to an outside human life. In rural areas, swiftmoving spooked wildlife such as deer,^{[41]} elk, moose, and antelope are more likely to intercept a roadway at over 30 mph (48 kph). Wildlife will frequently transit across a road before a full stop is necessary, however collisions with large game are foreseeably lethal, and a driver generally has a duty not to harm his or her passengers. The foreseeable intercept speed or defectively designed horizontal sight distance may vary "reasonably" with judicial discretion.
ACDA: intersectional setback
This third case regards safe speed around unsignalized intersections where a driver on an uncontrolled though street has a duty to slow down in crossing an intersection and permit controlled drivers to be able pass through the intersection without danger of collision.^{[35]} The driver on the through street must anticipate and hence not approach at an unsafe speed which would prevent another driver from being able to enter while traffic was some distance away, or would be unsafe to a driver who has already established control of the intersection under a prudent acceleration a_{i}, from a stop at a limit line a distance d_{sl} away.^{[38]}
ACDA: following distance
The pedantic fourth case applies when the dynamic ACDA "following distance" (d_{ACDAd}) is less than the static ACDA "lineofsight" distance (d_{ACDAs}). A classic instance of this occurs when, from a visibility perspective, it would be safe to drive much faster were it not for a slowermoving vehicle ahead. As such, the dynamic ACDA is governing the basic speed rule, because in maintaining this distance, one cannot drive at a faster speed than that matching the forward vehicle. The "time gap" t_{g} or "time cushion" is the time required to travel the dynamic ACDA or "following distance" at the operating speed. Circumstances depending, this cushion might be manifested as a twosecond rule or threesecond rule.
Critical speed
In the fifth case, critical speed V_{cs} applies when road curvature is the factor limiting safe speed. A vehicle which exceeds this speed will slide out of its lane. Critical speed is a function of curve radius "r," superelevation or banking "e," and friction coefficient "μ;"^{[32]} the constant "g" again is the acceleration of gravity. However, most motorists will not tolerate a lateral acceleration exceeding 0.3g (μ=0.3) above which many will panic.^{[43]} Hence, critical speed may not resemble loss of control speed.^{[43]} Attenuated "side" friction coefficients are often used for computing critical speed.^{[40]} The formula is frequently approximated without the denominator for low angle banking which may be suitable for nearly all situations except the tightest radius of highway onramps.^{[40]}^{[44]}
Surface control
The bottom case is invoked when the maximum velocity for surface control V_{cl} is otherwise reached. Steering control is independent from any concept of clear distance ahead. If a vehicle cannot be controlled so as to safely remain within its lane above a certain speed and circumstance, then it is irrelevant how assuredly clear the distance is ahead. Using the example of the previous case, the safe speed on a curve may be such that a driver experiences a lateral acceleration of less than 0.3g despite that the vehicle may not slide until it experiences 0.8g. Speed wobble, hydroplaning, roll center, fishtailing, jackknife tendencies, potholes, washboarding, and frost heaving^{[45]} are other factors limiting V_{cl}.
Safe speed
Safe speed is the maximum speed permitted by the basic speed law and negligence doctrine. Safe speed is not the same as the 85 percentile "operating"^{[46]} speed used by traffic engineers in establishing speed zones.^{[32]} Fog, snow, or ice can create conditions where most people drive too fast, and chain reaction accidents in such conditions are examples of where large groups of drivers collided because they failed to reduce speed for the conditions.^{[32]} The speeds at which most people drive can only be a very rough guide to safe speed,^{[32]} and an illegal or negligent custom or practice is not in itself excusable.^{[47]} Safe speed approximates the inferred design speed adjusted for environmental alterations and vehicle and person specific factors when V_{ACDAs} is the limiting factor.^{[48]}
Famous tragedies
Actor James Dean's 85 mph car crash, Princess Diana's death, and the sinking of the RMS Titanic after colliding with an iceberg at night are all well publicized tragedies which resulted from a failure to maintain an assured clear distance ahead.
"Assurance" beyond proximate edge of clear visibility as transference of liability
A general principle in liability doctrine is than an accident which would not have occurred except for the action or inaction of some person or entity contrary to a duty such as the exercise of proper care was the result of negligence. Jurisdictional exceptions permitting one to legally take "assurance" that the distance will be clear beyond the proximate edge of clear visibility and choose such a speed accordingly, transfers liability from that driver for his or her "blind" actions. This duty to assure clear distance ahead is most easily transferred to the government and its road engineers and maintainers. As it is generally foreseeable that, chance permits, at some point there will be an obstruction beyond some driver's line of sight, such an entitlement challenges established negligence doctrines in addition to posing difficult policy and engineering challenges.
To be able to guarantee this "assurance," a road must be designed and maintained such that there is not a chance of obstruction in one's lane beyond the proximate edge of clear visibility. A road's vertical profile must be assured to have such a curvature as not to hide hazards close behind its crests. Discretion for drivers and pedestrians to enter onto a potentially occupied lane from a side street must be assuredly eliminated such as with fences, merge lanes, or signalized access. There must also be an assurance of no opportunity for animals and debris to enter from side lots, and that there are continuous multihourly maintenance patrols performed. Furthermore, such road sections must be distinguished from other roads so that the driver could clearly and immediately know when he or she may or may not take such extended "assurance." Few roads might meet these requirements except some of the highest functional classification freeways and autobahns.
Even if such criteria are met, the law must also exempt driver liability for maintaining clear distance ahead. In most democracies, such liability for failures of the distance to remain clear beyond line of sight would ultimately be transferred to its taxpayers. This only generally occurs when governments have been tasked by constituents or their courts to take the responsibly to design and maintain roadways that "assure" the distance will be clear beyond the proximate edge of clear visibility. Pressures to make such changes may arise from cultural normalization of deviance and unnecessary risk, misunderstanding the purpose of the road functional classification system, underestimation of increased risk, and reclamation of commute time.
One of the greatest difficulties created by such an extension of the ACDA is the frequency at which roads reduce their functional classification unbeknownst to drivers who continue unaware they have lost this extended "assurance" or don't understand the difference. Such nuance in applicable jurisdictions is a prolific source of accidents.^{[33]} In the United States, there is no explicit road marking promising clear distance beyond line of sight in the Manual on Uniform Traffic Control Devices, although there are signs communicating "limited sight distance," "hill blocks view," "crossroad ahead," and "freeway ends."^{[49]} A partial solution to this challenge is to remove driver discretion in determining whether the ACDA is extended beyond line of sight, by explicitly designating this law change to certain marked high functional classification roadways having meet strict engineering criteria.
See also
Notes

^ In addition to being old common law principle, ACDA case law jurisprudence buttressed the legislature's implicit intent in its Basic Speed Law, and further limited transference of liability for ACDA negligence to the state—under the California Tort Claims Act—for insufficient sight distance at the speed of which the driver chose. See CVC § 22350, CVC § 22358.5, Cal Gov. Code § 830.4, Cal Gov. Code § 830.8, and Cal Gov. Code § 831. See CACI Form 1120 for details.

^ For this reason, full corner sight distance is almost never required for individual driveways in urban highdensity residential areas, and street parking is commonly permitted within the rightofway.

^ In most jurisdictions, judicial notice shall be taken of the total stopping distance, and such notice is therefore logically and substantively taken of the maximum speed permitted to brake within the stopping distance as applied to the ACDA. The latter is merely the inverse function of the former. Furthermore, fundamental mathematical relationships are themselves subject to judicial notice. V_{ACDA(s)}=\sqrt{\mu^2 g^2 t_{prt}^2+ 2 \mu g d_{ACDA_s} }  \mu g t_{prt} For example, using the \mu=0.7 and t_{prt}=1.5 values that produced Code of Virginia § 46.2880 Tables of speed and stopping distances, one simply obtains the same velocities that produced the stopping distance in the statute: Metric (SI) – Speed in km/h from distance in meters: V_{ACDA} \approx \sqrt{1372.3+ 177.8 d_{ACDA} }  37.0 US customary – Speed in MPH from distance in feet: V_{ACDA} \approx \sqrt{529.8+ 20.9 d_{ACDA} }  23.0

^ Safe Speed will be outputted in the same terms as the input units. Entering a distance in feet and an acceleration in terms of feet/s^{2} will produce a safe speed in terms of feet/second. To convert to miles per hour, multiply by 360/528. Entering distance and acceleration in terms of meters will output a speed in meters per second, which may be converted to kilometers per hour by multiplying a 3.60 factor.
References

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^ ^{a} ^{b} ^{c}

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^ See California Official Reports: Online Opinions

^ See California Official Reports: Online Opinions

^ ^{a} ^{b} See California Official Reports: Online Opinions

^ ^{a} ^{b} Driver traveling at 35 MPH when rain limited visibility to 25 feet held negligent when 65 feet were required to stop car on wet road. See California Official Reports: Online Opinions

^ ^{a} ^{b} See California Official Reports: Online Opinions

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^ ^{a} ^{b} ^{c} See Official Reports Opinions Online

^ ^{a} ^{b} ^{c} See Official Reports Opinions Online

^

^ ^{a} ^{b} ^{c} See Huetter v. Andrews, 91 Cal. App. 2d 142, Berlin v. Violett, 129 Cal.App. 337, Reaugh v. Cudahy Packing Co., 189 Cal. 335, and Official Reports Opinions Online

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^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g}

^ ^{a} ^{b}

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^ ^{a} ^{b} ^{c}

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^ ^{a} ^{b} ^{c} ^{d} ^{e}

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Further reading: tertiary sources
ACDA related law reviews
Other printed resources
Web resources

All about car accidents: The "Assured Clear Distance Ahead" Rule
Derivations
Case 1: Safe speed as a function of forward lineofsight
Derivation of ACDA 1
Forces on a vehicle skidding down a grade of angle θ.
Starting with Newton's Second Law of Motion and the Laws of Friction:

F_{total} = F_{friction} + F_{gravity} \sin{\theta}

F_{total} = \mu F_{normal} + m g \sin{\theta}

F_{total} = \mu m g \cos{\theta} + m g \sin{\theta}
Equating the net force to mass times acceleration:

F_{total}= m a

\mu m g \cos{\theta} + m g \sin{\theta} = m a

a = g(\mu \cos{\theta} + \sin{\theta})
Invoking the equations of motion and substituting acceleration:

d = \frac{v^2}{ 2 a}

d = \frac{v^2}{ 2 g(\mu \cos{\theta} + \sin{\theta})}
Smallangle approximation:

\sin{\theta} \approx \theta

\cos{\theta} \approx 1  \frac {\theta^2}{2}
Substituting the small angle approximations, and exploiting that the product of a small angle squared, in radians, with the friction coefficient, θ^{2}μ, is insignificant (for a steep 20% slope and a good friction coef of 0.8, this equals (.2)^{2}x0.8≈0.03):

d \approx \frac{v^2}{ 2 g[\mu (1  \frac {\theta^2}{2} ) + \theta]} \approx \frac{v^2}{ 2 g(\mu + \theta)}
Now, the total stopping distance is the sum of the breaking and perceptionreaction distances:

d_{total} = d_{breaking} + d_{perceptionreaction}

d_{total} \approx \frac{v^2}{ 2 g(\mu + \theta)} + v t_{pr}
Isolating zero as preparation to solve for velocity:

\frac{1}{2 g (\mu + \theta)} v^2 + v t_{prt}  d_{total} \approx 0
Completing the square or invoking the quadratic formula to find the solution:

v \approx \sqrt{(\mu + \theta)^2 g^2 t_{prt}^2+ 2 (\mu + \theta) g d_{total} }  ( \mu + \theta ) g t_{prt}
Use smallangle approximation to obtain a more fieldable version of the above solution in terms of percent grade/100 "e" instead of an angle θ in radians:

\theta \approx tan(\theta) = \frac{%grade}{100}
Substituting the angle as described produces the form of the formula of case 1 ():

V_{BSL1} \approx \sqrt{(\mu + e)^2 g^2 t_{prt}^2+ 2 (\mu + e) g d_{ACDA} }  (\mu+e) g t_{prt}
The Basic Speed Law constrains the Assured Clear Distance Ahead to thetotal stopping distance, and the small angle value of road grades approximates the superelevation "e."
Many roadways are level, in which case the small angle approximations or superelevation may be dropped altogether:

V_{BSL1} = \sqrt{\mu^2 g^2 t_{prt}^2+ 2 \mu g d_{ACDA} }  \mu g t_{prt}
Case 2: Safe speed as a function of horizontal lineofsight
Derivation of ACDA 2
The time required for an obstruction with speed v_{i} to transect the horizontal sight distance d_{i}:

t = \frac{d_i}{v_i}
The time required to travel down a road at speed v to said obstruction of distance d away:

t= \frac{d}{v}
Equating the two times:

\frac{d}{v} = \frac{d_i}{v_i}
Solving for this distance:

d=\frac{v d_{i}}{v_i}
Equating this to the total stopping distance, which is the sum of breaking and perceptionreaction distances:

\frac{v d_{i}}{v_i} = \frac{v^2}{2 g (\mu + e)} + v t_{prt}
Isolating zero, and factoring out a v:

v [ \frac{v}{2 g (\mu + e)} + (t_{prt}  \frac{ d_{i}}{v_i}) ] = 0
Solving for the nontrivial case (or may distribute v in equation above and apply quadratic formula for same result):

\frac{v}{2 g (\mu + e)} + (t_{prt}  \frac{ d_{i}}{v_i}) = 0
The solution to the above equation, which provides the maximum safe speed as a function of horizontal sight distance, intercept velocity, and roadtire friction coefficient:

v = 2 g (\mu + e) ( \frac{ d_{i}}{v_i}  t_{prt})
Case 3: Safe speed as a function of intersectional setback
Derivation of ACDA 3
The time required for a vehicle to enter from a controlled intersection is the sum of the perception time, the time required to actuate an automatic transmission or shift to first gear, and the time to accelerate and enter or traverse the road. The sum of the first two quantities is t_{pc}.

t= t_p + t_c + t_a = t_{pc} + t_a
The time required for a vehicle entering with acceleration a_{i} to transect the sum of the setback and shoulder distances d_{i} under uniform acceleration a_{i} from a stop via the equations of motion:

t_a =\sqrt{ \frac{ 2 d_i }{a_i} }
The time required to travel down a road at speed v to said obstruction of distance d away:

t= \frac{d}{v}
Equating the two times:

\frac{d}{v} =\sqrt{ \frac{ 2 d_i }{a_i} } + t_{pc}
Solving for this distance:

d = v ( \sqrt{ \frac{ 2 d_i }{a_i} } + t_{pc} )
Equating this to the total stopping distance, which is the sum of breaking and perceptionreaction distances:

v ( \sqrt{ \frac{ 2 d_i }{a_i} } + t_{pc} )= \frac{v^2}{2 g (\mu + e)} + v t_{prt}
Isolating zero, and factoring out a v:

v [ \frac{v}{2 g (\mu + e)} + (t_{prt}  \sqrt{ \frac{ 2 d_i }{a_i} }  t_{pc} ) ] = 0
Solving for the nontrivial case (or may distribute v in equation above and apply quadratic formula for same result):

\frac{v}{2 g (\mu + e)} + (t_{prt}  \sqrt{ \frac{ 2 d_i }{a_i} }  t_{pc} ) = 0
The solution to the above equation, which provides the maximum safe speed as a function of horizontal setback, intercept acceleration, and roadtire friction coefficient:

v = 2 g (\mu + e) ( \sqrt{ \frac{ 2 d_i }{a_i} } + t_{pc}  t_{prt})
Case 4: Safe speed as a function of following distance
Derivation of ACDA 4
From the equations of motion:

t_g = \frac{d}{v}
Isolating for speed:

v = \frac{d}{t_g}
Case 5: Safe speed as a function of critical speed
Derivation of ACDA 5
Forces on a vehicle skidding down a grade of angle θ.
Starting with Newton's Laws of Motion, the Laws of Friction, and Centripetal force:

F_{centripetal} \cos{ \theta } = F_{friction} + F_{gravity} \sin{\theta}
Substituting formulas for Centripetal force, frictional force, and gravitational force:

m \frac{v^2}{r} \cos{ \theta } = \mu F_{normal} + m g \sin{\theta}
The normal force is equal and opposite to the sum of the gravitational and centripetal components:

m \frac{v^2}{r} \cos{ \theta } = \mu (m g \cos{\theta} + m \frac{v^2}{r} \sin{ \theta } ) + m g \sin{\theta}
Isolate v terms:

\frac{v^2}{r} \cos{ \theta }  \mu \frac{v^2}{r} \sin{ \theta } = g (\mu \cos{\theta} + \sin{\theta} )
Then solve for v:

v^2 ( \cos{ \theta }  \mu \sin{ \theta } ) = g r (\mu \cos{\theta} + \sin{\theta} )
To obtain:

v = \sqrt { \frac{g r (\mu \cos{\theta} + \sin{\theta} ) } { \cos{ \theta }  \mu \sin{ \theta } } }
This is the full solution, however most corners are banked at less than 15 degrees (≈28% grade), so in such conditions, a fieldable small angle approximation may be used.
Substituting smallangle approximations sinθ≈θ, cos≈1θ^{2}/2:

v \approx \sqrt { \frac{g r [\mu (1 \frac {\theta^2} {2}) + \theta ] } { 1 \frac {\theta^2}{2}  \mu \theta } }
Exploit that a small angle squared, in radians, is insignificant by substituting θ^{2}≈0 which obtains the formula used in case 5 (also tanθ≈e):

v \approx \sqrt { \frac{g r (\mu + \theta ) } { 1  \mu \theta } } \approx \sqrt { \frac{g r (\mu + e ) } { 1  \mu e } }
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