Problem 1 (25 points)
Part (a) (10 points)
On a one-kilometre running track, 30 ultra-marathon runners are interval training by running one lap (i.e. one kilometre) at 12km/h, briskly walking a recovery lap at 6km/h, and then continuing to alternate between lap speeds of 12 km/h and 6km/h for a very long period of time. To help visualise the system, you can assume that the runners start along the track in such a way that each speed class (i.e. those going at 6km/h or those going at 12km/h) maintains nearly uniform spacing over the track. (Provide all the answers in 2 decimal places)
(i) (4 points) Determine the total flow (runners/h) of runners past any point on the track. (ii) (2 points) Determine the time-mean speed of the runners. (iii) (2 points) Determine the space-mean speed of the runners. (iv) (2 points) The coach is interested in observing them only during their run laps. Determine the flow (runners/h) of runners past the coach?
Part (b) (15 points)
The speed and density data collected for Avengers Street in Schofields is presented in the table below. (Provide all the answers in 3 decimal places)
Survey data for Avengers Street
speed range (km/h) | frequency | density (vehicles/km) |
0-10 | 29 | 112.5 |
10-20 | 22 | 97.5 |
20-30 | 31 | 82.5 |
30-40 | 23 | 67.5 |
40-50 | 30 | 52.5 |
50-60 | 21 | 37.5 |
60-70 | 27 | 22.5 |
70-80 | 17 | 7.5 |
(i) (2 points) Sketch the relationship between different elements of traffic flow theory and clearly mark the free flow speed, jam density, critical density, maximum flow (ii) (5 points) Calibrate a Greenshields model explaining the relationship between speed and density. (iii) (2 points) Determine the free flow speed and jam density (iv) (2 points) Determine the relationship between flow and density (v) (2 points) Determine the relationship between speed and flow (vi) (2 points) Determine the maximum flow and density at maximum flow
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