Moving observer: car counts on 800 m road

The moving observer method gives the flow $q$ and mean stream speed from counts made by a test car running with and against the stream.

Reading of the table: the vehicles she meets while driving southward are the northbound stream she is measuring ($N_s$); while driving northward, $N_o$ = cars that overtake her and $N_p$ = cars she overtakes. Section length 800 m, test speed 20 km/h on both runs.

Average counts (4 runs)

$$\begin{aligned} \bar N_s &= \frac{107+113+30+79}{4} = 82.25\\ \bar N_o &= \frac{10+25+15+18}{4} = 17.00, \quad \bar N_p = \frac{74+41+5+9}{4} = 32.25\\ y &= \bar N_o - \bar N_p = -15.25 \end{aligned}$$

Travel times

$$t_w = t_a = \frac{0.8}{20} = 0.04\ \text{h} = 2.4\ \text{min}$$

Traffic flow

$$q = \frac{N_s + y}{t_w + t_a} = \frac{82.25 + (-15.25)}{0.04 + 0.04} = 837.5\ \text{veh/h}$$

Journey time and speed of the stream

$$\bar t = t_w - \frac{y}{q} = 0.04 - \frac{(-15.25)}{837.5} = 0.0582\ \text{h} = 3.49\ \text{min}$$ $$V_j = \frac{L}{\bar t} = \frac{0.8}{0.0582} = 13.74\ \text{km/h}$$

Answer: average flow = 837.5 veh/h; journey speed = 13.7 km/h (journey time 3.49 min). The stream is slower than the test car because more vehicles were overtaken by her than overtook her.