iPad '5' 8:43 AM 74% -' < icIoud Ch: Iii m 121 344 mm 74.1. Assume :40?) = 11...,(1 171/111\") and p.\" x<0 p(x,0) = {p.mu x)[l 0 <1: L. Determine p(x, t). 74.3. Consider the following partial di'erential equation: %p'%=0 --ool. 77.1. If u = Hana - p/pm.), then what is the velocity of a tratiic shock separat- ing densities p. and p, ? (Simplify the exptemion as much as possible.) Show that the shock velocity is the average of the density wave velocities associated with pa and p,. 71.2. If u =- u,\" (l p'lpgn), then what is the velocity of a traic shock separating densities p. and pl? (Simplify the expression as much as pos- sible.) Show that the shock velocity is not the average of the density wave velocities associated with po and p.. 78.1. Suppose that the initial tra'ic density is p x<0 p(x.0)= p1 x>0. Consider the two cases, po < p. and p, < pg. For which of the preceding cases is a density shock necessary? Briey explain. 18.2. Assume that u = u...(l pipnn) and that the initial traic density is 95! x<0 5 p(x,0)= ng-E x>0. (a) Sketch the initial density. (b) Determine and sketch the density at later times. (c) Determine the path of a car (in space-time) which starts at x = -xo (behind it = 0). (d) Determine the path of a car (in space-time) which starts at x = x, (ahead of x = O). iPad "3 < iCloud E? 74% -' (stoma 8:43 AM Assume that u = 11...,(1 - p/p...,.) and at t = 0, the trafc density is 33" x < 0 not. 0) = 2 '\"" x > 0. Why does the density not change in time? Referring to the problem in Sec. 78, show algebraically that the value of the shock velocity is between the velocities of the two density waves. Suppose that only a sketch of the ow-density curve is known. Assume that the density is known at t - 0. 8? x<0 3 "\" x>0. p(x. 0) = Determine the trafc density at later times. If u = u..,(l gamma, then at what velocity do cars pile up at a red traic light, assuming that the initial trafc density is a constant ,0... ms. Assume that not) :-u...;(l hype\"). Determine the traic density p (for r > 0) if the initial trafc density is p; x<0 so = \"x ) p3 x>0. (a) Assume that p; > m. (b) Assume thstp30 .o ={ \"(x ) o x<0. Determine the velocity of the shock. Briey give a physical explanation of the result. The initial trafc density on a road is 0 x g 0 p(x,0)= fe-x 0