Due to roadwork, the traffic on a highway slows linearly from 70 miles/hour to 5

Question

Due to roadwork, the traffic on a highway slows linearly from 70 miles/hour to 5 miles/hour over 2000 foot stretch of road, then crawls along at 5 miles/hour for 4500feet then speeds back up linearly to 70 miles/hour in the next 1750 feet, after which it moves steadily at 70 miles/hour. (a) Sketch a velocity vector field for the traffic flow. Then write a formula for the velocity vector v (in mi/hr) as a function of the distance x feet along the road from the initial point of slowdown. Take the direction of motion to be i and consider the various sections of the road separately. (b) Find each of the following values of the divergence. For each, include units. div v at x=1000: div v= ? div v at x=2500: div v=? div v at x=7750: div v= ?

Explanation / Answer

b.

0<x<2000 v = (70-x/(400/13))i

2000<x<6500 v = 5i

6500<x<8250 v = (5+(x-6500)/(350/13))i

x>8250 v = 70i

c.

x=1000 divV = -1/(400/13)mph/ft

x=2500 divV = 0 mph/ft

x=7750 divV = 1/(350/13) mph/ft

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