Question

On a road, the speed – density relationship of a traffic stream is given by \( u = 70 - 0.7k \) where speed, u, is in km/h and density, k, is in veh/km.  At the capacity condition, the average time headway will be:

• A. 0.5 s
• B. 1.0 s
• C. 1.6 s
• D. 2.1 s

Hint:

At the capacity condition, the time headway is the reciprocal of the flow. To maximize flow, differentiate the flow equation with respect to density and solve for the corresponding density.

Answer 1

The Correct Option is D

At the capacity condition, the traffic density \( k_c \) is given by the point where the flow is maximized, which occurs when the derivative of the flow with respect to density is zero. The flow \( q \) is given by: \[ q = u \times k = (70 - 0.7k) \times k. \] The maximum flow occurs at the density \( k_c \) that maximizes this expression. After differentiating and solving for \( k_c \), we find the density at capacity and can use it to find the average time headway \( t_h \), which is the inverse of flow. The average time headway \( t_h \) is given by: \[ t_h = \frac{1}{q}. \] At capacity, we find that the average time headway is 2.1 seconds. Final Answer: \[ \boxed{2.1 \, \text{seconds}}. \]