Mobility & Transport Networks

Plan de Formación (40h)

Plan de Formación (40h)
(2021)

Lecture 2. Trajectories
 Trajectories (s-t) + queues (N-t)
Transport operations:
• Trajectories (in movement –
overcome the distance)
– Speed, acceleration, etc.
s
• Queues (waiting – overcome
the time)
̶ Waiting time, maximum
queue and average, etc.
• Dual analysis of
movement

N
Q
t

t 2
 Definition

• Determination of the position of a vehicle (x) in relation to a

reference point during the time (t).

• Geometrical place of the vehicle position (space) respect to

the time: function x(t). Graphic representation of x(t) in the
plane (t, x) is a curve named trajectory.

• Applications:
– Minimum travel time between stations in a railway line for a
maximum speed.
– Determine the initial speed of a car from the braking marks on its
tires.
– Length of runways and location of exits.
– Length of acceleration / deceleration lanes.
– Etc.

3
 Diagrams s-t
x

* t
(*) does not represent a real physic al trajectory

TRAJECTORY
Instantaneous speed = v(t) = dx(t)/dt
Instantaneous acceleration = a(t) = dv(t)/dt = d2x(t)/dt2
In case, a(t) = a(constant):

4
 Interpretación de trayectorias (1/2)
s t Acceleration
(recovering
Deceleration speed)
(intersection,
pedestrian
crossing, bump)

Free
speed

Vehicle running at a
lower speed and with
a longer than the
previous vehicles

5
 Interpretación de trayectorias (2/2)

¿If we record a video

in a section of a road,
what do we see?

6
 Construction of Trajectories
A) by record times of vehicle passing through fixed sections (elevators,
pedestrians, automobiles, etc.).
B) by photographs in sequenced areas in time or by time ranges in systems
operated by scheduled in a circuit (buses with navigation systems).
C) by record of the overtaking time of vehicles to an observer in movement at a
constant speed.
D) by continuous “tracking” with GPS (in vehicles such as cars, trucks, urban
buses via SAE or private smart phones) or bluetooth (roads).

7
Micro and Macro traffic variables

Micro: h, headway

Macro: q, flow

i=1

Micro: s, spacing

Macro: k, density

j=1

8
Diagrams (s,t) in traffic

Real trajectories in
a highway

9
Traffic jams in highways

10
Comparison of PT modes

11
 Bus bunching

TRAJECTORIES D'AUTOBUSOS

4500,00 bus 1
4000,00 bus 2
3500,00 bus 3
TEMPS (seg.)

3000,00 bus 4
2500,00 bus 5
2000,00 bus 6
1500,00 bus 7
1000,00 bus 8
500,00 bus 9
0,00 bus 10
0 2 4 6 8 10 12 bus 11
PARADES bus 12

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 Bus bunching
AUTOBUSES PAREADOS

2000

1800

1600

1400

1200
Tiem po (s)

1000

800

600

400

200

0
0 5 10 15 20 25
Paradas

13
 Bus bunching
AUTOBUSES PAREADOS

2000

1800

1600

1400

1200
Tiem po (s)

1000

800

600

400

200

0
0 5 10 15 20 25
Paradas

14
 Bus bunching
AUTOBUSES PAREADOS

2000

1800

1600

1400

1200
Tiem po (s)

1000

800

600

400

200

0
0 5 10 15 20 25
Paradas

15
 Bus bunching
AUTOBUSES PAREADOS

2000

1800

1600

1400

1200
Tiem po (s)

1000

800

600

400

200

0
0 5 10 15 20 25
Paradas

16
 Bus bunching
AUTOBUSES PAREADOS

2000

1800

1600

1400

1200
Tiem po (s)

1000

800

600

400

200

0
0 5 10 15 20 25
Paradas

17
 Bus bunching
AUTOBUSES PAREADOS

2000

1800

1600

1400

1200
Tiem po (s)

1000

800

600

400

200

0
0 5 10 15 20 25
Paradas

18
 Bus bunching
AUTOBUSES PAREADOS

2000

1800

1600

1400

1200
Tiem po (s)

1000

800

600

400

200

0
0 5 10 15 20 25
Paradas

19
 Trajectories PV and PT

Trajectòries
4.000

3.800

3.600

3.400
TRAJECTORIES B
TRAJECTORY PV OF BUSES
3.200
H
3.000

2.800
A
2.600 E
2.400
G
Posició

2.200

2.000 C
1.800 I
1.600
H2
1.400

1.200 A2
1.000
J
800

600
B2
400

200

0
0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Temps

20
 Group speed
3 friends are travelling with a tandem bicycle. The bicycle travels at 20 km/h and
walking the speed is 4 km/h at a constant rate. The operative is:
• A+B in tandem, C walking
• After a while (t), B walks and A goes back and picks up C with the bicycle
• A+C by bike until they find B.... A walks and B+C by bike... Repetition of the cycle

Average group speed?

21
 Group speed
3 friends are travelling with a tandem bicycle. The bicycle travels at 20 km/h and
walking the speed is 4 km/h at a constant rate. The operative is:
• A+B in tandem, C walking
• After a while (t), B walks and A goes back and picks up C with the bicycle
• A+C by bike until they find B.... A walks and B+C by bike... Repetition of the cycle

Average group speed? x

4
1
2t
20(t  x)  4(t  x)  x 
3 1 20 t
20
20
4  (5t / 3)  20  (t ) 80t / 3 1
v   10km / h
8t / 3 8t / 3
20
1
1
4
20t/3
time
t (2/3)t t (8/3)t
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 Group speed
Bike speed = b km/h; walking speed = w km/h
(w/b<1).
x
w
1

1
b bt
b
D
1

b
1
w w(t+y)
1

t y t
t
t
T
If b = 20 km/h, w = 4 km/h,  = 1/2 (Vgrupo = 0,5b = 10 km/h).
If b = 20 km/h, w = 20 km/h,  = 1 (Vgrupo = b = 20 km/h).
If b = 20 km/h, w = 0 km/h,  = 1/3 (Vgrupo = 0,33b = 6,67 km/h).
If b = 30 km/h, w = 4 km/h,  = 0,4468 (Vgrupo = 13,404 km/h).
If b = 10 km/h, w = 4 km/h,  = 0,6471 (Vgrupo = 6,47 km/h). Was it “intuitive”?
If b = 20 km/h, w = 5 km/h,  = 0,5385 (Vgrupo = 10,769 km/h). 23
 Group speed

4 10-15
b(bici)= w= w= w= w= w= w=
2 5-10
0 1 2 3 4 5
0 0 0 0 0 0 0
0-5
1 0,333333 1 1,4 1,666667 1,857143 2 0
2 0,666667 1,428571 2 2,444444 2,8 3,090909 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
3 1 1,8 2,454545 3 3,461538 3,857143
4 1,333333 2,153846 2,857143 3,466667 4 4,470588
5 1,666667 2,5 3,235294 3,888889 4,473684 5
6 2 2,842105 3,6 4,285714 4,909091 5,478261
7 2,333333 3,181818 3,956522 4,666667 5,32 5,923077 16
8 2,666667 3,52 4,307692 5,037037 5,714286 6,344828
9 3 3,857143 4,655172 5,4 6,096774 6,75
14
10 3,333333 4,193548 5 5,757576 6,470588 7,142857
11 3,666667 4,529412 5,342857 6,111111 6,837838 7,526316
12 4 4,864865 5,684211 6,461538 7,2 7,902439 12 14-16
13 4,333333 5,2 6,02439 6,809524 7,55814 8,272727
14 4,666667 5,534884 6,363636 7,155556 7,913043 8,638298 10 12-14
15 5 5,869565 6,702128 7,5 8,265306 9 10-12
16 5,333333 6,204082 7,04 7,843137 8,615385 9,358491 8
17 5,666667 6,538462 7,377358 8,185185 8,963636 9,714286 8-10
18 6 6,872727 7,714286 8,526316 9,310345 10,0678 6
19 6,333333 7,206897 8,050847 8,866667 9,655738 10,41935 6-8
20 6,666667 7,540984 8,387097 9,206349 10 10,76923 4 4-6
21 7 7,875 8,723077 9,545455 10,34328 11,11765
22 7,333333 8,208955 9,058824 9,884058 10,68571 11,46479 2 2-4
23 7,666667 8,542857 9,394366 10,22222 11,0274 11,81081
24 8 8,876712 9,72973 10,56 11,36842 12,15584 0 0-2
25 8,333333 9,210526 10,06494 10,89744 11,70886 12,5 0
26 8,666667 9,544304 10,4 11,23457 12,04878 12,84337
2 4 6 8 10
27 9 9,878049 10,73494 11,57143 12,38824 13,18605 12 14 16 18 20 0
28 9,333333 10,21176 11,06977 11,90805 12,72727 13,52809 22 24 26 28
29 9,666667 10,54545 11,40449 12,24444 13,06593 13,86957 30
30 10 10,87912 11,73913 12,58065 13,40426 14,21053

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 Playful fly

2 trains separated by an initial distance D moving in opposite directions

with speeds v1 y v2 until crash. A fly is travelling between train headboards
with a (high and constant) speed u (u >> max{v1, v2})

How far has the fly travelled before dying in the

crash?

25
 Playful fly
2 trains separated by an initial distance D moving in opposite directions
with speeds v1 y v2 until crash. A fly is travelling between train headboards
with a (high and constant) speed u (u >> max{v1, v2})

How far has the fly travelled before dying in the

crash?

x
x Dx D
v1  v2  t
D t t v1  v2
1
v2 Distance travelled  t·u

Du
x t u 
v1  v2
u v1
1 1

0
t
t

26