The speed-density relationship in a mid-block section of a highway follows the Greenshield's model. If the free flow speed is $v_{f}$ and the jam density is $k_{j}$ , the maximum flow observed on this section is

\frac{v_{f} k_{j}}{2}

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\frac{v_{f} k_{j}}{8}

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v_{f} k_{j}

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\frac{v_{f} k_{j}}{4}

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Solution

The correct option is D $\frac{v_{f} k_{j}}{4}$
$Q = v_{f} [k - \frac{k^{2}}{k_{j}}]$ .....(i)
For maximum Q,
$\frac{d Q}{d k} = 0$
$v_{f} [1 - \frac{2 k}{k_{j}}] = 0$
$\Rightarrow k = \frac{k_{j}}{2}$
putting value of k in equation (i)
$t h e r e f o r e Q = \frac{k_{j}}{2} \times \frac{v_{f}}{2} = \frac{v_{f} k_{j}}{4}$

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