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Geometric Design: Theory and Concepts

Braking Distance

The braking distance is the distance that a vehicle travels while slowing to a complete stop. The braking distance is a function of several variables. First, the slope (grade) of the roadway will affect the braking distance. If you are going uphill, gravity assists you in your attempts to stop and reduces the braking distance. Similarly, gravity works against you when you are descending and will increase your braking distance. Next, the frictional resistance between the roadway and your tires can influence your braking distance. If you have old tires on a wet road, chances are you'll require more distance to stop than if you have new tires on a dry road. The last parameter that we will consider is your initial velocity. Obviously, the higher your speed the longer it will take you to stop, given a constant deceleration. 

The equation used to calculate the braking distance is a child of a more general equation from classical mechanics. The parent equation is given below.


Vf = Final velocity
Vo= Initial velocity
a = Acceleration rate
d = Distance traversed during acceleration

When calculating the braking distance, we assume the final velocity will be zero. Based on this, the equation can be manipulated to solve for the distance traversed during braking.

d = -Vo2/(2a) 

Notice that the distance will be positive as long as a negative acceleration rate is used.

The acceleration of a braking vehicle depends on the frictional resistance and the grade of the road. From our knowledge of the frictional force, we know that the acceleration due to friction can be calculated by multiplying the coefficient of friction by the acceleration due to gravity.  Similarly, we know from inclined plane problems that a portion of the car's weight will act in a direction parallel to the surface of the road. The acceleration due to gravity multiplied by the grade of the road will give us an estimate of the acceleration caused by the slope of the road. 

The final formula for the braking distance is given below.  Notice how the acceleration rate is calculated by multiplying the acceleration due to gravity by the sum of the coefficient of friction and grade of the road.

d = V2/(2g(f + G))

d = Braking Distance (ft)
g = Acceleration due to gravity (32.2 ft/sec2)
G = Roadway grade as a percentage; for 2% use 0.02
V = Initial vehicle speed (ft/sec)
f = Coefficient of friction between the tires and the roadway

The braking distance and the brake reaction time are both essential parts of the stopping sight distance calculations. In order to ensure that the stopping sight distance provided is adequate, we need a more in-depth understanding of the frictional force. The value of the coefficient of friction is a difficult thing to determine. The frictional force between your tires and the roadway is highly variable and depends on the tire pressure, tire composition, and tread type. The frictional force also depends on the condition of the pavement surface. The presence of moisture, mud, snow, or ice can greatly reduce the frictional force that is stopping you. In addition, the coefficient of friction is lower at higher speeds. Since the coefficient of friction for wet pavement is lower than the coefficient of friction for dry pavement, the wet pavement conditions are used in the stopping sight distance calculations. This provides a reasonable margin of safety, regardless of the roadway surface conditions. The table below gives a few values for the frictional coefficient under wet roadway surface conditions (AASHTO, 1984).

Design Speed (mph) Coefficient of Friction (f)
20 0.40
30 0.35
40 0.32
60 0.29