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 1150 lb to large trucks weighing 7440 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.

Answer 1

The speed limit in an area (55 mph) is too high to allow vehicles to stop in time.

v = 55 mph

converting mph into ft/s :

v = 24.5 ft/s

Since friction is the force slowing down the car.

Then, we have

F = m a f = m a

_s m g = m a

_s g = a                                                                    { eq.1 }

A maximum acceleration will be given by -

a_max = [(0.480) (32 ft/s²)]

a_max = 15.3 ft/s²

Then, the minimum braking distance which will be given by -

d_min = v² / 2 a_max

d_min = (24.5 ft/s)² / [2 (15.3 ft/s²)]

d_min = [(600.25 ft²/s²) / (30.6 ft/s²)]

d_min = 19.6 ft

A minimum acceleration will be given by -

a_min = [(0.350) (32 ft/s²)]

a_min = 11.2 ft/s²

Then, the maximum braking distance which will be given by -

d_max = v² / 2 a_min

d_max = (24.5 ft/s)² / [2 (11.2 ft/s²)]

d_max = [(600.25 ft²/s²) / (22.4 ft/s²)]

d_max = 26.7 ft