Either the question is wrong or I'm missing something obvious

[pasmith]

Homework Statement

This is exercise 2.4.2 from Pringle and King, Astrophysical Flows (CUP 2007):

Along an infinite, straight, one-track road the local density of cars is $\rho(x,t)$ and the local velocity of cars (all assumed to be traveling in the same direction) is $v(x,t)$. Discuss why it might be reasonable to assume that $v$ is solely a function of $\rho$.

Making this assumption, show that $$\frac{\partial \rho}{\partial t} + c(\rho) \frac{\partial \rho}{\partial x} = 0$$ where the kinematic wave speed is defined by $c(\rho) = Q'(\rho)$ and $Q = \rho v$ is the local flux of cars.

Traffic flow along a particular highway can be fitted approximately for $\rho < \rho_{\mathrm{max}}$ by $$Q(\rho) = V_0\rho \log(\rho_{\mathrm{max}}/\rho),$$ where $V_0 = 25\,\mathrm{kph}$ and $\rho_{\mathrm{max}} = 150\,\mathrm{vechicles}\,\mathrm{km}^{-1}$.

Show that information propagates upstream at a speed $V_0$ relative to the local vehicle velocity.

Show that there is a maximum traffic flow which occurs at some density $\rho_{\mathrm{crit}}$ corresponding to a critical speed $v_{\mathrm{crit}}$ of around 75 kph.

[Remainder omitted]

Homework Equations

The Attempt at a Solution

All is straightforward until we consider the particular choice of $Q$. Firstly we have $$c = Q' = V_0\left(\log(\rho_{\mathrm{max}}/\rho) - 1\right) = \frac Q\rho - V_0 = v - V_0$$ so indeed $c - v = -V_0$.

Now we're asked to find the maximum traffic flow. This corresponds to the maximum of $Q$, and we've just shown that $Q' = v - V_0$. Thus the maximum occurs at $v_{\mathrm{crit}} = V_0$, at which point $\rho_{\mathrm{crit}} = e^{-1}\rho_{\mathrm{max}}$.

According to the question, $v_{\mathrm{crit}} \approx 3V_0$. Am I missing something obvious, or is the question wrong?

 

[D H]

You apparently had no problem deriving the kinematic wave equation (you didn't ask for help on that part), but then you promptly ignored that "waviness" in the rest of your analysis. You need to account for the fact that traffic flows in waves, at least when density is high enough.

You found the maximum instantaneous flux. It's the traffic flux $\bar Q$ averaged over time that you should be maximizing rather than that instantaneous flux.