Question

On a specific highway, the speed-density relationship follows the Greenberg's model$[v=v_{f}lo{g}_e(k_j/k\left)\right]$, where $v_f$ and $k_j$ are the free flow speed and jam density respectively. When the highway is operating at capacity, the density obtained as per this model is

• A. $e{k}_j$
• B. $\frac{k_j}{2}$
• C. $\frac{k_j}{e}$
• D. $k_j$
  

Answer 1

The correct option is C $\frac{k_j}{e}$

Velocity, $V=V_{f}ln\left(\frac{k_j}{k}\right)$
$V_f$ and $k_j$ are the free flow speed and jam density, capacity =density $\times$ speed,
$q=k.V_{f}ln\left(\frac{k_j}{k}\right)$
For maximum flow,$\frac{dq}{dk}=0$
$\Rightarrow k.V_f.\left(\frac{k}{k_j}\right).k_j(-\frac{1}{k^2})+V_{f}ln\left(\frac{k_j}{k}\right)$
$-V_f+V_{f}ln\frac{k_j}{k}=0$
$ln\frac{k_j}{k}=1=lne$
Density,
$k=\left(\frac{k_j}{e}\right)$