C E 34500: Transportation Engineering
Spring 2020
H om ew ork 2
1. Two sets of students are collecting traffic data at two sections, xx and yy, of a highway 1500
ft. apart. Observations at xx show that five vehicles passed that section at intervals of 3, 4,
3, and 5 sec, respectively. If the speeds of the vehicles were 50, 45, 38, 35, and 30 mi/hr
respectively, draw a schematic showing the locations of the vehicles 20 sec after the first
vehicle passed section xx. Also determine (a) the time mean speed, (b) the space mean
speed, and (c) the density on the highway.
Answer:
The distance traversed by each vehicle 20 seconds after crossing section x-x is calculated as
follows:
Vehicle A: 𝑋𝐴 = 50 ∗ 1.47 ∗ 20 = 1470 𝑓𝑡.
Vehicle B: 𝑋𝐵 = 45 ∗ 1.47 ∗ (20 − 3) = 1125 𝑓𝑡.
Vehicle C: 𝑋𝐶 = 38 ∗ 1.47 ∗ (20 − 3 − 4) = 726 𝑓𝑡.
Vehicle D: 𝑋𝐷 = 35 ∗ 1.47 ∗ (20 − 3 − 4 − 3) = 515 𝑓𝑡.
Vehicle E: 𝑋𝐸 = 30 ∗ 1.47 ∗ (20 − 3 − 4 − 3 − 5) = 221 𝑓𝑡.
X-X
a) Time mean speed=
E
D
C
B
A
221’
515’
726’
1125’
1470’
50+45+38+35+30
5
b) Space mean speed =
5
1
1 1 1 1
+ + + +
50 45 38 35 30
5 𝑣𝑒ℎ
3600
∗
3+4+3+5 38.3
c) Density =
Y-Y
= 39.6 mph
= 38.3 mph
= 31.3 𝑣𝑒ℎ/𝑚𝑖
Page 1 of 6
2. The following dataset consists of 30 observations of vehicle speed and length taken from a 6ft by 6-ft inductive loop detector during a 60-second time period. Determine the occupancy,
density, and flow rate.
Vehicle
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Speed, mph
61
66
62
70
65
69
72
66
65
64
67
68
65
66
71
64
59
58
64
64
68
58
66
57
64
61
69
63
63
66
Length, ft.
18
17
19
21
16
26
21
19
20
20
25
70
35
20
65
24
23
22
65
30
24
21
56
21
20
50
19
23
17
18
Answer:
Vehicle
Speed, mph, u
Length, ft., L
1
2
3
4
5
6
7
8
9
10
11
61
66
62
70
65
69
72
66
65
64
67
18
17
19
21
16
26
21
19
20
20
25
Time Vehicle Spends
in Detector Zone
0.27
0.24
0.27
0.26
0.23
0.32
0.26
0.26
0.27
0.28
0.31
Page 2 of 6
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
68
65
66
71
64
59
58
64
64
68
58
66
57
64
61
69
63
63
66
70
35
20
65
24
23
22
65
30
24
21
56
21
20
50
19
23
17
18
0.76
0.43
0.27
0.68
0.32
0.33
0.33
0.75
0.38
0.30
0.32
0.64
0.32
0.28
0.62
0.25
0.31
0.25
0.25
10.75
Total=
Average
Occupancy =
Density=
10.75
60
= 17.92%
𝑂𝑐𝑐𝑢𝑝𝑎𝑛𝑐𝑦
𝐿𝑚𝑒𝑎𝑛 +𝑑
Flowrate= 𝑛 ∗
28.2
3600
𝑇
0.1792
= .28.2+6 = 0.0052
= 30 ∗
3600
60
𝑣𝑒ℎ
𝑓𝑡
= 1800
= 27.66 𝑣𝑒ℎ/𝑚𝑖𝑙𝑒 (assuming similar lengths of vehicles)
𝑣𝑒ℎ
ℎ𝑟
3. The data shown below were obtained by time-lapse photography on a highway. Use
regression analysis to fit these data to the Greenshields model and determine (a) the mean
free speed, (b) the jam density, (c) the capacity, and (d) the speed at maximum flow.
Speed, mph
14.2
24.1
30.3
40.1
50.6
55
Density, veh/mi
85
70
55
41
20
15
Answer:
Page 3 of 6
70
60
Speed
50
40
Speed = -0.5684*Density + 62.813
R² = 0.9965
30
20
10
0
0
20
40
60
80
100
120
Density
When density =0; Speed=62.8= Free flow speed
When speed=0, density=110.5 = jam density
Speed at max flow= 62.8/2= 31.4
Capacity=62.8 ∗
110
4
= 1727 𝑣𝑒ℎ/ℎ𝑟
4. Traffic on the eastbound approach of a signalized intersection is traveling at 35 mi/hr, with
a density of 46 veh/mi/ln. The duration of the red signal indication for this approach is 30
sec. If the saturation flow is 1900 veh/h/ln with a density of 52 veh/mi/ln, and the jam
density is 125 veh/mi/ln, determine the following:
(i) The length of the queue at the end of the red phase
(ii) The maximum queue length
(iii) The time it takes for the queue to dissipate after the end of the red indication.
Answer:
Page 4 of 6
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5. If a traffic stream is modeled using Greenshields model, prove that the space mean speed at
which the volume is maximum is equal to half the free mean speed. Similarly, the density at
which the volume is maximum is equal to half the jam density.
Answer:
Page 6 of 6