Question: The velocity field v→ in Problem does not give a complete description of the traffic flow, for it takes no account of the spacing between vehicles. Let ρ be the density (cars/mile) of highway, where we assume that ρ depends only on x.
(a) Using your highway experience, arrange in ascending order: ρ(0), ρ(1000), ρ(5000).
(b) What are the units and interpretation of the vector field ρv→?
(c) Would you expect ρv→ to be constant? Why? What does this mean for div(ρv→)?
(d) Determine ρ(x) if ρ(0) = 75 cars/mile and ρv→ is constant.
(e) If the highway has two lanes, find the approximate number of feet between cars at x = 0, 1000, and 5000.
Problem: Due to roadwork ahead, the traffic on a highway slows linearly from 55 miles/hour to 15 miles/hour over a 2000-foot stretch of road, then crawls along at 15 miles/hour for 5000 feet, then speeds back up linearly to 55 miles/hour in the next 1000 feet, after which it moves steadily at 55 miles/hour.
(a) Sketch a velocity vector field for the traffic flow.
(b) Write a formula for the velocity vector field v→ (miles/hour) as a function of the distance x feet from the initial point of slowdown. (Take the direction of motion to be i and consider the various sections of the road separately.)
(c) Compute div v→ at x = 1000, 5000, 7500, 10,000. Be sure to include the proper units.