Instantaneous Stream Measurement — Flow, Density and the Two Mean Speeds: 16-Civ-B10 Traffic Engineering, December 2014, Q1

Question 1: Instantaneous Stream Measurement — Flow, Density and the Two Mean Speeds (5 + 15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A single instant-of-time (“aerial photograph”) observation of one direction of travel over a 100 m length of a two-lane highway. Four vehicles are present, each 6 m long, positioned by the distance of the rear of the vehicle from a west reference point, with flow running west to east.

Vehicle	Rear of vehicle from west reference point (m)	Front of vehicle (m)	Spot speed (km/h)
A (leading)	75	81	72
B	40	46	64
C	20	26	56
D (trailing)	5	11	32
Section length observed, L		100 m = 0.100 km
Vehicle length		6 m each

Find. (a) a dimensioned sketch of the four vehicles drawn to scale on the 100 m section, and (b) the flow q, the density k, the time-mean speed u_t and the space-mean speed u_s of the stream at that instant.

Part (a) — dimensioned sketch of the instantaneous vehicle positions. Each vehicle is drawn to scale as a 6 m rectangle with its rear face at the tabulated distance from the west reference point; the clear gaps between successive vehicles (29 m, 14 m and 9 m) and the overall 100 m section length are dimensioned.

Approach. A snapshot over a known length of road measures density directly, so obtain k from the vehicle count, obtain the two mean speeds from the spot speeds (arithmetic mean for the time-mean, harmonic mean for the space-mean), and then recover flow from the fundamental relation of traffic flow, which requires the space-mean speed.

Part (a) — fix the vehicle geometry before drawing. Each rectangle runs from its tabulated rear position to a point 6 m further downstream (east), so the front faces sit at 81, 46, 26 and 11 m. The clear gap between two successive vehicles is the rear of the leading vehicle less the front of the following one: $$\text{gap}_{\text{A-B}} = 75 - 46 = 29\ \text{m},\qquad \text{gap}_{\text{B-C}} = 40 - 26 = 14\ \text{m},\qquad \text{gap}_{\text{C-D}} = 20 - 11 = 9\ \text{m}$$ The gaps close down towards the rear of the platoon while the speeds also fall from 72 to 32 km/h, which is exactly the behaviour a speed–spacing relation predicts and is worth noting on the sketch. The dimensioned sketch is drawn above.
Part (b) — density from the vehicle count in the known length. Density is the number of vehicles occupying a unit length of road, measured here directly: $$k = \frac{n}{L} = \frac{4\ \text{veh}}{0.100\ \text{km}} = \boxed{40\ \text{veh/km}}$$ As a check, the mean spacing over the section is $L/n = 100/4 = 25\ \text{m}$, and $k = 1000/25 = 40\ \text{veh/km}$, which agrees.
Time-mean speed is the arithmetic mean of the spot speeds. The time-mean speed is the average of the speeds of vehicles passing a point, and with a single observation of each vehicle it is simply $$u_t = \frac{1}{n}\sum_{i=1}^{n} u_i = \frac{72 + 64 + 56 + 32}{4} = \frac{224}{4} = 56.00\ \text{km/h}$$
Space-mean speed is the harmonic mean of the same speeds. The space-mean speed is the length of road divided by the average travel time over it, which makes it a harmonic rather than an arithmetic average: $$u_s = \frac{n}{\displaystyle\sum_{i=1}^{n} \frac{1}{u_i}} = \frac{4}{\frac{1}{72} + \frac{1}{64} + \frac{1}{56} + \frac{1}{32}} = \frac{4}{0.078621} = 50.88\ \text{km/h}$$ The individual reciprocals are 0.013889, 0.015625, 0.017857 and 0.031250 h/km. Note that $u_s < u_t$, as it must be for any stream in which the speeds are not all identical — the slow vehicles occupy the section for longer and therefore carry more weight in a space average.
Flow follows from the fundamental relation, using the space-mean speed. The generalised relation of traffic flow is $q = k\,u_s$, and it is valid only with the space-mean speed: $$q = k\,u_s = 40\ \text{veh/km} \times 50.88\ \text{km/h} = \boxed{2035\ \text{veh/h}}$$ Had the time-mean speed been used by mistake, the answer would have been $40 \times 56.00 = 2240\ \text{veh/h}$ — an overstatement of about 10 %, which is precisely the error the two-mean-speed distinction exists to prevent.

Check: 2035 veh/h is above the roughly 1900–2000 pc/h/lane that a two-lane highway lane can actually sustain. That is not an arithmetic error — it is a property of the measurement. A single 100 m snapshot containing four vehicles is a very small sample of a very short length, so the density (and therefore the flow) it implies is a local, instantaneous estimate and must not be read as a sustainable hourly volume. Reporting it as “the flow at that instant” is the honest statement; a design volume would require a full volume study over 15-minute or hourly intervals.

Quantity	Symbol	Value
Density	k	40.0 veh/km
Time-mean speed	u_t	56.00 km/h
Space-mean speed	u_s	50.88 km/h
Flow	q = k u_s	2035 veh/h
Average spacing	L/n	25.0 m
Clear gaps (A–B, B–C, C–D)	—	29 m, 14 m, 9 m

Topic: the three macroscopic stream variables and why there are two mean speeds.

Key relations: $k = n/L$; $u_t = \frac{1}{n}\sum u_i$; $u_s = n / \sum (1/u_i)$; the fundamental relation of traffic flow $q = k\,u_s$; and the approximate identity $u_t \approx u_s + \sigma_s^2/u_s$ linking the two means through the speed variance.

Why this works: flow, density and speed are not three independent measurements; any two fix the third. A measurement made over a length of road at one instant gives density directly, whereas a measurement made at a point over an interval of time gives flow directly. The two mean speeds arise from those two sampling frames. A time average at a point counts each vehicle once regardless of how long it stays in view, so fast vehicles are over-represented relative to their share of the road space; a space average weights each vehicle by its occupancy time, which is why it is the harmonic mean and why it is always the smaller of the two. Only the space-mean speed multiplies with density to give flow, because density is itself a space measure — mixing a time average with a space measure produces a systematically high flow.

Common pitfall: computing $q = k\,u_t$ (here 2240 instead of 2035 veh/h), and measuring the vehicle positions to the front of each rectangle when the table specifies the rear, which shifts every gap by 6 m and corrupts the sketch dimensions.

Source: Garber & Hoel, Traffic and Highway Engineering, 5th ed., Ch. 6 §6.1–6.2 (traffic flow elements; time-mean and space-mean speed).

Question 1 of 6: Instantaneous Stream Measurement — Flow, Density and the Two Mean Speeds

Question 1: Instantaneous Stream Measurement — Flow, Density and the Two Mean Speeds (5 + 15 marks)