Question

A single-lane highway has a traffic density of 40 vehicles/km. If the time-mean speed and space-mean speed are 40 kmph and 30 kmph, respectively, the average headway (in seconds) between the vehicles is:

• A. 3.00
• B. 2.25
• C. \(8.33 \times 10^{-4}\)
• D. \(6.25 \times 10^{-4}\)

Hint:

The average headway is the reciprocal of the traffic density, and the relationship between time-mean speed and space-mean speed is crucial in determining traffic flow and headway.

Answer 1

The Correct Option is A

The relationship between the traffic density \( k \), space-mean speed \( v_s \), and time-mean speed \( v_t \) is given by: \[ k = \frac{1}{h} \] where \( h \) is the headway (time between vehicles) in seconds. We know: \[ k = 40 \, \text{vehicles/km}, \quad v_t = 40 \, \text{km/h}, \quad v_s = 30 \, \text{km/h}. \] The time-mean speed \( v_t \) is related to the space-mean speed \( v_s \) by the equation: \[ v_t = \frac{v_s}{1 + k \cdot h} \] Substitute the known values: \[ 40 = \frac{30}{1 + 40 \cdot h}. \] Solving for \( h \): \[ 1 + 40h = \frac{30}{40}, \quad 40h = \frac{30}{40} - 1 = \frac{30 - 40}{40} = \frac{-10}{40}. \] Hence, \[ h = 3 \, \text{seconds}. \] Thus, the correct answer is (A).