# 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.

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.