As a city planner, you receive complaints from local residents about the safety

Question

As a city planner, you receive complaints from local residents about the safety of hearby 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 feet. 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 0.842 and 0.941, whereas the effective coefficient of friction between a skidding (locked) wheel and asphalt ranges between 0.550 and 0.754. Vehicles of all types travel on the road, from small VW bugs weighing 1310 lb to large trucks weighing 8670 lb. Considering that some drivers will brake properly when slowing down and others will skid to stop, calculate the miminim and maximum braking distance needed to ensure that all vehicles traveling at the posted speed limit can stop before reaching the intersection. 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.

Explanation / Answer

posted speed=55 mph=24.5872 m/s

as we know,

final speed^2-initial speed^2=2*acceleration*distance

taking initial speed=24.5872 m/s

final speed=0 m/s

and as acceleration is opposing to the motion, it will be having -ve sign .

let magnitude of deceleration is a.

stopping distance=24.5872^2/(2*a)

hence greater the deceleration, lesser the stopping distance

deceleration=friction coefficient*acceleration due to gravity

==>deceleration=friction coefficient*9.8 m/s^2

hence greater the friction coefficient , greater the deceleration and lesser the stopping distance

for rolling friction, coefficients are from 0.842 and 0.941:

coefficient: 0.842==> breaking distance=24.5872^2/(2*0.842*9.8)=36.631 m

coefficient : 0.941==>breaking distance=24.5872^2/(2*0.941*9.8)=32.777 m

for skidding, friction coefficient ranges from 0.55 to 0.754

coefficient: 0.55==>breaking distance=24.5872^2/(2*0.55*9.8)=56.0788 m

coefficient: 0.754 ==> breaking distance=24.5872^2/(2*0.754*9.8)=40.9063 m

hence minimum distance is 32.777 m=107.536 ft

maximum distance=56.0788 m=183.9855 ft

part 2:

maximum breaking distance occurs in case of lowest coefficient.

so if fix the speed limit with respect to this friction coefficient, for the same speed, higher coefficientsd will have lesser breaking distance

given that maximum breaking distance=155 ft=47.244 m

minimum coefficient of friction=0.55

then maximum speed limit=sqrt(2*acceleration*breaking distance)

=sqrt(2*coefficient of friction*9.8*47.244)

=sqrt(2*0.55*9.8*47.244)=22.5674 m/s=50.482 mph

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