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 1070 lb to large trucks weighing 8550 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? Precipitation from the fog can lower the coefficients of friction. Newton's second law does not apply to this situation. Drivers cannot be expected to obey the posted speed limit. Reaction time of the drivers is not taken into account.

Answer 1

Given

Rolling friction coefficient range is = 0.689 -0.770

Skidding friction range is 0.450 - 0.617

Visibility length = 155 ft = 47.244 m

Initial speed u = 55 mph = 24.585 m/s

Solution

F = ma

Since in our case friction is the one force acting on the car

μmg = ma

a = μg

Minimum breaking distance

When the deceleration is at its maximum then the distance taken to come to stop would be minimum.

Since acceleration is proportional to μ. we should consider the maximum μ value for this out of all given range

a_max = 0.770 g

a_max = 0.770 x 9.8

a_max = 7.546 m/s²

V² = u²- 2as

0 = 24.585² - 2 x 7.546 x s_mim

S_mim = 40.05 m = 131.4 ft

Maximum breaking distance

When the deceleration is at its minimum then the distance taken to come to stop would be maximum.

Since acceleration is proportional to μ. we should consider the minimum μ value for this out of all given range

a_min = 0.450 g

a_min = 0.450 x 9.8

a_min = 4.41 m/s²

V² = u²- 2as

0 = 24.585² - 2 x 7.4.41x s_max

S_max = 58.53 m = 185.47 ft

Desired Speed limit

The desired speed limit should allow everyone to stop before the intersection safely, even if they have minimum acceleration. Since the visibility is limited to 155 ft (47.244 m)

a_min= 4.41 m/s

V² = u² - 2as

0 = u² - 2 x 4.41 x 47.244

u = 20.4 m/s = 45.6 miles per hour

So speed limit should be 45 miles per hour