Question

Traffic on a northbound highway segment is stationary and is composed of two families of vehicles; cars which travel at speed vc and trucks which travel at speed vt . A moving observer traveling in the opposing (i.e., southbound) direction travels a distance L in time T, L / T = v o. During this trip, the moving observer counts the number of (northbound) cars, mc, and the number of (northbound) trucks, mt, that pass her.

Derive an expression for the total flow of vehicles in the northbound direction in terms of the variables given.

Answer 1

Consider a stream of vehicles moving in the north bound direction. Two different cases of motion can be considered. The first case considers the traffic stream to be moving and the observer to be stationary.

Figure 1: Illustration of moving observer method

If n_o is the number of vehicles overtaking the observer during a period, t, then flow q is , or

(1)

The second case assumes that the stream is stationary and the observer moves with speed v_o. If n_p is the number of vehicles overtaken by observer over a length l, then by definition, density k is , or

(2)

or

(3)

where v₀ is the speed of the observer and t is the time taken for the observer to cover the road stretch. Now consider the case when the observer is moving within the stream. In that case m_o vehicles will overtake the observer and m_p vehicles will be overtaken by the observer in the test vehicle. Let the difference m is given by m₀ - m_p, then from equation 1 and equation 3,

(4)

This equation is the basic equation of moving observer method, which relates q,k to the counts m, t and v_o that can be obtained from the test. However, we have two unknowns, q and k, but only one equation. For generating another equation, the test vehicle is run twice once with the traffic stream and another one against traffic stream, i.e.

where, a,w denotes against and with traffic flow. It may be noted that the sign of equation 6 is negative, because test vehicle moving in the opposite direction can be considered as a case when the test vehicle is moving in the stream with negative velocity. Further, in this case, all the vehicles will be overtaking, since it is moving with negative speed. In other words, when the test vehicle moves in the opposite direction, the observer simply counts the number of vehicles in the opposite direction. Adding equation 5 and 6, we will get the first parameter of the stream, namely the flow(q) as:

(7)

Now calculating space mean speed from equation 5,

If v_s is the mean stream speed, then average travel time is given by t_avg = . Therefore,

Rewriting the above equation, we get the second parameter of the traffic flow, namely the mean speed v_s and can be written as,

(8)

Thus two parameters of the stream can be determined. Knowing the two parameters the third parameter of traffic flow density (k) can be found out as

(9)

For increase accuracy and reliability, the test is performed a number of times and the average results are to be taken.