Question

As a city planner, you receive complaints from local residents about the safety of nearby roads and streets. One complaint concerns a stop sign at the corner of Pine Street and 1st Street. Residents complain that the speed limit in the area (55 mph) is too high to allow vehicles to stop in time. Under normal conditions this is not a problem, but when fog rolls in visibility can reduce to only 155 ft. Since fog is a common occurrence in this region, you decide to investigate.

The state highway department states that the effective coefficient of friction between a rolling wheel and asphalt ranges between 0.689 and 0.770, whereas the effective coefficient of friction between a skidding (locked) wheel and asphalt ranges between 0.450 and 0.617.Vehicles of all types travel on the road, from small VW bugs weighing 1430 lb to large trucks weighing 9180 lb.

Considering that some drivers will brake properly when slowing down and others will skid to stop, calculate the minimum and maximum braking distance needed to ensure that all vehicles traveling at the posted speed limit can stop before reaching the intersection.

minimum braking distance: ft

maximum braking distance: ft

Given that the goal is to allow all vehicles to come safely to a stop before reaching the intersection, calculate the maximum desired speed limit.

maximum speed limit: mph

Which factors affect the soundness of your decision?

Drivers cannot be expected to obey the posted speed limit.

Reaction time of the drivers is not taken into account.

Precipitation from the fog can lower the coefficients of friction.

Newton's second law does not apply to this situation.

Answer 1

While applying break, retardation given to the vehicle = g , where is the friction coefficient between rolling wheel and asphalt and g is acceleration due to gravity.

For a speed limit 55 mph ( 24.6 m/s ) , breaking distance i.e. required distance d to stop vehicle after applying break

is calculated from , u² = 2 a d   , where u is initial speed at the time of applying break, a is retardation and d is distance traveled after applying break.

hence we get, d = u²/ ( 2 g ) = { 24.6 24.6 / ( 2 9.8 ) } m = ( 30.875 / ) m

To get distance in feet, d = ( 101 / ) ft

type of wheel	Coefficient of friction	Distance required to stop after applying break ( ft )
Rolling	0.69	147
Rolling	0.77	132
Skidding	0.45	225
Skidding	0.617	164

Hence minimum breaking distance = 132 ft ; Maximum breaking distance = 225 ft

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Now we find the safe speed limit, by considering the distance 155 ft 47 m as the distance required to stop the vehicle after applying break

As explained above, we use the relation, u² = ( 2 a d ) to get speed limit using d = 47 m and retardation a = g

Hence safe speed limit, u = 30.35 m/s = 67.9 mph

Type of wheel	Coefficient of friction	speed limit ( mph )
rolling	0.69	56
rolling	0.77	59.6
skidding	0.45	45.5
skidding	0.617	53.3

From above table , Safe speed limit = 45 mph