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 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 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 0.536 and 0.599, whereas the effective coefficient of friction between a skidding (locked) wheel and asphalt ranges between 0.350 and 0.480. Vehicles of all types travel on the road, from small VW bugs weighing 1310 lb to large trucks weighing 7.80 x10 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. Minimum Maximum Number Number 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. (Scroll down for more questions.) Number mph

Explanation / Answer

55 mph 80.7 ft/s

trucks:
Ek = ½mv² = ½ * (7800lb / 32.2 ft/s²) * (80.7ft/s)² = 788779ft·lb
worst case friction: Ffw = µmg = 0.55 * 7800lb = 4290 lb
stopping distance d = Ek / Ffw = 183 ft
best case friction: Ffb = 0.599 * 7800lb = 4672 lb
stopping distance d = Ek / Ffb = 168 ft

bugs:
Ek = ½ * (1310lb / 32.2ft/s²) * (80.7ft/s)² = 137 354 ft·lb
worst case friction: Ffw = 0.55 * 1310lb = 720lb
stopping distance d = 191 ft
best case friction: Ffb = 0.599 * 1310lb = 785 lb
stopping distance d = 175 ft

Given that the maximum allowable distance is 155 ft, we've got to reduce the maximum allowable Ek of the vehicles, and it appears not to matter which one we analyze.
worst case friction for bug over 155 ft entails Work = 720lb * 155ft = 111600ft·lb
This corresponds to Ek = 111600ft·lb = ½ * (1310lb / 32.2ft/s²) * v²
v 74 ft/s 50 mph maximum desired speed limit

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