Question

The free mean speed is 60 km/hr on a given road. The average space headway at jam density on this road is 8 m. For a linear speed-density relationship, the maximum flow (in veh/hr/lane) expected on the road is:

• A. 1875
• B. 938
• C. 2075
• D. 1038

Hint:

Remember, the critical point in a linear speed-density relationship occurs at half the jam density, maximizing flow at this density under ideal linear conditions.

Answer 1

The Correct Option is A

Step 1: Determine the jam density. Jam density \( k_j \) is given by the inverse of the space headway at jam density: \[ k_j = \frac{1}{\text{space headway}} = \frac{1}{8 \text{ m}} = 125 \text{ vehicles/km}. \] Step 2: Apply the Greenshields’ model for linear speed-density relationship to find the critical density. The critical density \( k_c \) where flow is maximum is half of the jam density: \[ k_c = \frac{k_j}{2} = \frac{125}{2} = 62.5 \text{ vehicles/km}. \] Step 3: Calculate the maximum flow using the formula \( q_{\text{max}} = k_c \times v_c \), where \( v_c \) is the speed at critical density. Using the linear relationship \( v = v_f \left(1 - \frac{k}{k_j}\right) \), where \( v_f \) is the free flow speed: \[ v_c = v_f \left(1 - \frac{k_c}{k_j}\right) = 60 \text{ km/hr} \left(1 - \frac{62.5}{125}\right) = 30 \text{ km/hr}. \] Then, maximum flow is: \[ q_{\text{max}} = k_c \times v_c = 62.5 \text{ vehicles/km} \times 30 \text{ km/hr} = 1875 \text{ vehicles/hr/lane}. \]