CVEN4404 Online Moodle Quiz 1 Solutions PDF

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Three cars are travelling on a 1 km long circular track at a speed of 40 km/h, 50 km/h and 60 km/h respectively. Assume you are an observer standing at a point on the track for a period of 6 minutes and are recording the instantaneous speed of each vehicle as it crosses your point.
a) Determin...

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School of Civil and Environmental Engineering Term 1, 2021

CVEN4404 FUNDAMENTALS OF TRAFFIC ENGINEERING

Moodle Quiz Solutions Question 1(1a - 1d) Speed and density data are collected for a lane of highway which are shown in the following table: Speed (km/h) 53.2 48.1 44.8

Density (veh/km) 20 27 35

40.1 37.3 35.2 34.1 27.2 20.4 17.5

44 52 58 60 64 70 75

14.6 13.1 11.2 8

82 90 100 115

Using linear regression analysis, calibrate the parameters of the Greenshields model explaining the relationship between speed and density Greenshields model: 𝑣𝑠 = 𝑣𝑓 −

𝑣𝑓 𝑘𝑗

𝑘

For simplicity, let’s use 𝑣𝑠 = 𝑎 + 𝑏𝑘 where 𝑎 = 𝑣𝑓 and 𝑏 = −

𝑣𝑓 𝑘𝑗

. 𝑛 ∑ 𝑘𝑣 −∑ 𝑘 ∑ 𝑣

Minimizing the sum of squared errors and setting the derivatives to 0, we have 𝑏 = 𝑛 ∑ 𝑘𝑠2−(∑ 𝑘)2 𝑠 and 𝑎 = 𝑣𝑠 − 𝑏𝑘 . Feeding the data into the equations results in 𝑏 = −0.528 and 𝑎 = 62.556. Hence the Greenshields model is 𝒗𝒔 = 𝟔𝟐. 𝟓𝟓𝟔 − 𝟎. 𝟓𝟐𝟖𝒌.

a) Determine the free flow speed CVEN4404 – Term 1 2021 – Course Profile Page 1

Based on the previous calculations, 𝒗𝒇 = 𝟔𝟐. 𝟓𝟓𝟔 km/h b) Determine the jam density 𝒗𝒇 𝒌𝒋 = 𝟎.𝟓𝟐𝟖 = 𝟏𝟏𝟖. 𝟒𝟕𝟕 veh/km. c) Determine the density at maximum flow. Use the flow-density relationship 𝑞 = 62.556𝑘 − 0.528𝑘2 . 𝑑𝑞 = 0, 62.556 − 0.2 ∗ 0.528𝑘 = 0 𝑑𝑘

𝟔𝟐.𝟓𝟓𝟔

The density at maximum flow is − 𝟐(−𝟎.𝟓𝟐𝟖) = 𝟓𝟗. 𝟐𝟑𝟗 veh/km d) Determine the maximum flow. Use the flow-density relationship 𝑞 = 62.556𝑘 − 0.528𝑘2 . The maximum flow is 1852.866 veh/h.

Question 2(2.a - 2d) Three cars are travelling on a 1 km long circular track at a speed of 40 km/h, 50 km/h and 60 km/h respectively. Assume you are an observer standing at a point on the track for a period of 6 minutes and are recording the instantaneous speed of each vehicle as it crosses your point. a) Determine density of the traffic on the track There are three vehicles on the track, and the track is one kilometre in length Density = number of vehicles per unit length = 3 vehicles / 1 km The Density is 3 vehicles per kilometre b) Determine the observed flow (in veh/h)? In the time period of 6 minutes (1/10 of an hour), the 40 km/h vehicle would pass the point 4 times, the 50 km/h vehicle would pass the point 5 times and the 60 km/h vehicle would pass the point 6 times. That is, a total of 4+5+6 = 15 vehicle passes would be observed. Flow = number of vehicles per time = 15 vehicles / 6 minutes = 2.5 vehicles per minute = 2.5 × 60 = 150 vehicle per hour The Hourly Volume is 150 vehicles per hour c) Determine space mean speed. Show the calculations for (i) using flow-density relationship and (ii) using spot speeds.

(i) using flow-density relationship Flow = Density × Space Mean Speed 150 veh/h = 3 veh/km × Space Mean Speed (km/h) Space Mean Speed = 150 ÷ 3 The Space Mean Speed is 50 kilometres per hour

(ii) using spot speeds

CVEN4404 – Term 1 2021 – Moodle Quiz Solutions Page 2

Harmonic mean means the reciprocal of the average of the reciprocals. 𝑣𝑠 =

1 ∑𝑛𝑖=1

1 𝑣𝑖

In the time period of 6 minutes (1/10 of an hour), the 40 km/h vehicle would pass the point 4 times, the 50 km/h vehicle would pass the point 5 times and the 60 km/h vehicle would pass the point 6 times. That is, a total of 4+5+6 = 15 vehicle passes would be observed. Thus, n = 15. 𝑣𝑠 =

1 15 = = 50 1 1 1 0.3 4∗ + 5 ∗ + 6 ∗ 10 50 60

The space mean speed calculated as the harmonic mean of spot speeds is 50 kilometres per hour

d) Determine time mean speed. Show the calculations for (i) using spot speeds and (ii)using space mean speed and the variance of the spot speeds.

(i) using spot speeds In the time period of 6 minutes (1/10 of an hour), the 40 km/h vehicle would pass the point 4 times, the 50 km/h vehicle would pass the point 5 times and the 60 km/h vehicle would pass the point 6 times. The time mean speed is the numerical average of the speeds of these 15 vehicles. 𝟒 ∗ 𝟒𝟎 + 𝟓 ∗ 𝟓𝟎 + 𝟔 ∗ 𝟔𝟎 𝟕𝟕𝟎  = 𝒗_𝒕 = = 𝟓𝟏. 𝟑 𝟒+𝟓+𝟔 𝟏𝟓 The time mean speed calculated as the numerical mean of spot speeds is 51.3 kilometres per hour (ii) using space mean speed and the variance of the spot speeds. The time mean speed can be calculated from the space mean speed and the variance of the spot speeds from the space mean speed using the following formula:

𝑣𝑡 = 𝑣𝑠 +

𝜎𝑠2 =

𝜎𝑠2 where 𝑣𝑠

𝜎𝑠2 =

∑𝑛 ) 𝑠 𝑖 (𝑣𝑖−𝑣

4∗(40−50) 2+5∗(50− 50)2 +6∗(60−50)2 4+5+6

𝜎2

66.6

𝑠

50

𝑠 = 50 + 𝑣𝑡 = 𝑣𝑠 +  𝑣

𝑛

=

100 15

= 66.6

= 50 + 1.3 = 51.3

The time mean speed calculated as the numerical mean of spot speeds is 51.3 kilometres per hour

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