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 (89 km/h) 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 47 meters. 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 with a mass of 627 kg to large trucks with mass 3447 kg. 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

89 km/h = 55.30 mph = 81.10 ft/s

trucks:
Ek = ½mv² = ½ * (7600lb / 32.2 ft/s²) * (81.10 ft/s)² = 776193 ft·lb
worst case friction: Ffw = µmg = 0.55 * 7600lb = 4180 lb
stopping distance d = Ek / Ffw = 186 ft
best case friction: Ffb = 0.941 * 7600lb = 7152 lb
stopping distance d = Ek / Ffb = 108 ft = 33m

bugs:
Ek = ½ * (1383 lb / 32.2ft/s²) * (81.10ft/s)² = 140899 ft·lb
worst case friction: Ffw = 0.55 *1383lb = 760 lb
stopping distance d = 186 ft
best case friction: Ffb = 0.941 *1383lb = 1301 lb
stopping distance d = 108 ft = 33 m

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 = 760lb * 155ft = 117800 ft·lb
This corresponds to Ek = 117800 ft·lb = ½ * (1383lb / 32.2ft/s²) * v²
v 74 ft/s 50 mph maximum desired speed limit

= 0.05 km/h ans

note

i will change all the kg to lb and i also use feet

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