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GATE 2025 CE (CE2) – Question 50

Traffic Engineering and Transportation Planning · Fundamental relationships of traffic flow · 2 marks · Multiple select

The free flow speed of a highway is 100 km/h and its capacity is 4000 vehicle/h. Assume speed density relation is linear.
For a traffic volume of 2000 vehicle/h, choose all the possible speeds (in km/h) from the options given below (round off to two decimal places).

  1. 85.36
  2. 65.20
  3. 14.64
  4. 7.22

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Show answer and explanation

Correct answer: (A) 85.36; (C) 14.64

Explanation

With a linear speed-density relation (Greenshields), $v = v_f\left(1 - \frac{k}{k_j}\right)$ and the capacity is $q_{max} = \frac{v_fk_j}{4}$, so $k_j = \frac{4 \times 4000}{100} = 160$ veh/km. For $q = 2000$: $100k\left(1 - \frac{k}{160}\right) = 2000$, so $0.625k^2 - 100k + 2000 = 0$ and $k = \frac{100 \pm 70.71}{1.25} = 23.43$ or $136.57$ veh/km. The speeds are $100\left(1 - \frac{23.43}{160}\right) = 85.36$ km/h (uncongested) and $100\left(1 - \frac{136.57}{160}\right) = 14.64$ km/h (congested).