Consider the green light problem from class and let X(t) be the position at time t of the car which starts a distance X0 from the light. With the linear velocity density function we found the car begins to move at time T1=X0/um and that in the fan X satisfies the initial value problem

dtdX2t1X=21um,X(T1)=X0

(a) Solve this first order linear equation to find X(t) in the fan

(b) Find, in terms of T1 only, the time T2 at which the car reaches the location of the light (x=0)

(c) Find the speed of the car in the limit t

(d) What happens to the distance between adjacent cars as t

(e) Use the easy method to find (x,t) in the fan for the green light problem with the quadratic velocity density function u()=um(12/m2)

Optional:

(f) Show directly that (x,t)=m[1L+2umtumt+x] satisfies t+q()x=0 with the quadratic velocity density function q()=um(1/m)

(Hint: It's easier if you write (x,t)=2m[1+L+2umtL2x]

(g) Using the quadratic velocity density function in problem (e), find (x,t) if (x,0)=m(1x/L).

Your answer should agree with problem (e) if L=0