Solve the Riemann Problem

Consider the Riemann Problem (RP) for the traffic flow model

$u_t + (u − u^2)_x = 0$
$U_L = 5/6$ and $U_R = 1/2$

i) Solve the RP. It will have a rarefaction wave.

ii) In the model,

$(u − u^2)/u = 1 − u$

(flux divided by density) is the scaled average speed of an individual car. Find the formula for the position $x(t)$ of a car that starts at $x(0) = −1$ in the traffic described by the RP above. Hint: it will be a three part solution, with two easy parts.