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Time/space mean speed and density from spot speeds

Short answer

Time mean speed is the arithmetic mean, 57.90 km/h. Space mean speed is the harmonic mean, 56.71 km/h. With mean headway 4.5 s, flow is 800 veh/h, giving density 14.1 veh/km.

The following speed data were collected during a two-minute segment of a spot speed study (speed in kmph).

45, 55, 48, 52, 60, 48, 60, 42, 52, 65, 64, 63, 58, 56, 68, 54, 68, 64, 66, 70

Calculate: (i) The time mean speed; (ii) The space mean speed. What will be the average density of the above traffic stream if the mean headway is 4.5 sec?

(i) Time mean speed

The arithmetic mean of the 20 spot speeds:

vˉt=∑vin=115820=57.90 km/h\bar v_t = \frac{\sum v_i}{n} = \frac{1158}{20} = 57.90\ \text{km/h}

(ii) Space mean speed

The harmonic mean of the spot speeds:

vˉs=n∑1/vi=200.35265=56.71 km/h\bar v_s = \frac{n}{\sum 1/v_i} = \frac{20}{0.35265} = 56.71\ \text{km/h}

(iii) Average density

Mean headway 4.5 s gives the flow q=3600/4.5=800q = 3600/4.5 = 800 veh/h. From q=k vˉsq = k\,\bar v_s:

k=qvˉs=80056.71=14.11 veh/kmk = \frac{q}{\bar v_s} = \frac{800}{56.71} = 14.11\ \text{veh/km}

Answer: time mean speed = 57.90 km/h; space mean speed = 56.71 km/h; density = 14.1 veh/km.

✓ Answer checked by Claude Opus 5.5 (high effort)

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