This is a pretty simple question but I can't seem to understand it conceptually. The question is:
If the traffic flow is increasing as $x$ increases ($\frac{\partial q}{\partial x}>0$), explain physically why the density must be decreasing in time ($\frac{\partial \rho}{\partial t}<0$). Relavent equation $$\frac{\partial q}{\partial x} + \frac{\partial \rho}{\partial t} = 0$$
I understand that as $x$ decreases then traffic density would increase because $$\rho = \frac{1}{x+L}$$ where $L$ is length of the car. And that in turn would increase traffic flow because $$q = \rho (velocity)$$
But if $x$ increases, then $\rho$ would decrease and $q$ would decrease....I am obviously thinking of this problem in the wrong way because I am not getting what they want. Any ideas?
The relevant equation you quote shows that if $\frac{\partial q}{\partial x} \gt 0$, then $ \frac{\partial \rho}{\partial t} \lt 0$ This must be for cars moving toward $+x.$ The physical explanation I would give would center on the fact that if the flow increases with $x$, you imagine a length of road there are less cars coming in from the left than are leaving to the right. The number of cars in the length must therefore be decreasing.