GATE 2026 CE (CE2) – Question 60

Traffic Engineering and Transportation Planning · Fundamental relationships of traffic flow · 2 marks · Numerical answer

For a given traffic stream, the speed-density relationship is given as:

$$v = v_o \ln\left(\frac{k_j}{k}\right)$$

where $v$ is the mean speed (in km/h), and $k$ is the density (in veh/km).

Considering $v_o$ as 45 km/h, and $k_j$ as 200 veh/km, the maximum flow (in veh/h) for the given stream is __________ (*rounded off to the nearest integer*).

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Correct answer: 3311

Explanation

The flow is $q = kv = v_o k \ln\left(\frac{k_j}{k}\right)$. Setting $\frac{dq}{dk} = v_o\left[\ln\frac{k_j}{k} - 1\right] = 0$ gives $k = \frac{k_j}{e} = \frac{200}{2.718} = 73.58$ veh/km, where the speed is $v_o = 45$ km/h. The maximum flow is $q_{max} = \frac{v_o k_j}{e} = \frac{45 \times 200}{2.718} = 3311$ veh/h.