exploring engineering chapter 13 engineering kinematics
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Exploring Engineering
Chapter 13
Engineering Kinematics
Kinematics and Traffic Flow
Transportation
Transportation The movement of goods and people
Transportation Engineering Design and operation of facilities and
vehicles to enhance transportation
Topics Covered
Speed and acceleration
Kinematics: the relationships between distance, velocity/
speed, acceleration,and time
Highway capacity
Speed and Acceleration Speed is not the same thing as “velocity.” Both have
dimensions of distance/time but “speed” is only the
magnitude of “velocity” In one-dimensional problems, they are equivalent In multidimensional problems, “velocity” connotes direction as well
as “speed”; here we will only deal in one-dimensional problems The average speed is calculated as Δx/Δt, where Δx is the
distance traveled in time Δt. Acceleration has dimensions of speed/time, and the average
acceleration can be calculated as ΔV/Δt. It is the “slope” of a speed vs. time graph.
To calculate the instantaneous speed or acceleration, the “Δ”s used in these equations should be very, very small.Speed units:US (miles per hour)Rest of the world (kilometers per hour)SI units: meters per second (m/s)
Example A student sees his/her bus coming down the street
and starts running at 2.5 m/s toward the bus stop. The bus is traveling at 10.0 m/s. The student starts running toward the bus stop when the bus is 50. m behind. What is the maximum distance the student can be from the bus stop to catch the bus?
Know: A) Student runs at 2.5 m/s. B) Bus is traveling at 10.0 m/sc. C) Distance between them 50. m
Find: How far can the student be from the bus stop?
Solution How: In time Δt the bus travels a distance Δxb= Vb
Δt, and the student travels a distance Δxs= Vb Δt.
If they both reach the bus stop at the same time, then the bus will have traveled a distance 50. meters more than the student travels, or Δxb= Δxs+ 50 .m
Since Δxb= Vb Δt and Vb Δt then Vb Δt = Vb Δt + 50. m, or (Vb - Vs)(Δt) = 50. m
Solve: Solving for Δt we get
Δt = 50./(Vb - Vs) = 50./(10. – 2.5) = 6.7 seconds
Distance vs. Time chart
Distance vs. Time
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6 7 8
Time in seconds
Dis
tan
ce i
n m
eter
s
Student Bus
The slope of the line is the speed.
Average Speed Average speed is a measure of the
distance traveled in a given period of time. Suppose that during your trip to school, you
traveled a distance of 5.0 miles and the trip lasted 0.20 hours (12. minutes). The average speed of your car could be determined as
On the average, your car was moving with a speed of 25. miles per hour. During your trip, there may have been times that you were stopped and other times that your speedometer was reading 50. miles per hour; yet on the average you were moving with a speed of 25. miles per hour.
miles/hr.25hours 0.20
miles 0.5Speed Ave.
Kinematics of Motion
Simplified case: constant acceleration
x = Vot+ (1/2)at2 (a = acceleration)
V= dx/dt (or Δx/ Δt, slope of distance vs. time curve)
So, V= Vo + at
a = dV/dt (or ΔV/Δt, slope of velocity vs time curve)
So, a = (dV/dx)(dx/dt) = (dV/dx)V
We will not be using these physics results
“Equations of motion”
Graphical View
The various regions in this graph represent Stationary, Constant speed, and Variable speed (i.e., acceleration)
Origin
Dis
tanc
e
Time
Stationary
Constant velocity
Variable velocity
Origin
Dis
tanc
e
Time
Stationary
Constant velocity
Variable velocity
speed
speed
Constant Velocity Algebraic Method
Apply Cartesian
geometry
Note: speed is distance/time Dimensions are
length/time Typical units ft/s, m/s
0
0
tt
xxV
)tV(txx 00
Origin
Dis
tanc
e
Time
x0
t0t
x
Constant VelocityCalculus Method
Speed is the rate of change of distance with time
We can integrate from the initial conditions (x0, t0) since V is constant
)t(txx constant tx 00 VV
Graphical Method
Consider the various regions in this graph Constant
speed Constant
acceleration Variable
acceleration
Origin
Spe
ed
Time
Constant speed, V0
Constant acceleration
Variable acceleration
Constant Acceleration Algebraic Method
0
0
tt
VVa
00 tta VV
00
00
21
21 Also,
ttaVV
VVVV
aver
aver
OriginS
peed
Time
Constant speed, V0
t0 t
V
Vaver
Use of speed/time diagram We have already
seen that the slope of the speed vs. time is acceleration
The area under the speed/time curve between t and to is the distance covered in that time interval
OriginS
peed
Time
Constant speed, V0
t0 t
V
Vaver
ExampleYou are designing an automated highway using vehicle speeds of 100. mph. How long does the on-ramp need to be to allow the car to reach this speed and how long will it take the vehicle to accelerate to this speed? Assume the vehicle will start at V0 = 0 and a = 5.00 ft/s2 at t = 0 sec.
Example Continued
Need: x at V = 100. mph and t at 100
mph. Note: 100 mph
= 5280 ×100./3600. [ft/mile] [miles/hr]
[hr/s] = 147 ft/s
Know: a = 5.00 ft/s2, V0= 0 ft/s at t0 = 0 s
How? Use a speed time graph
1) V/t = a or t = 147/5.00 = 29.4 s
gives the time
2) Area of triangle = ½ 147 29.4
[ft/s] [s] = 2160 ft gives distance
0t sec
Spe
ed, f
t/s
a =
5.00
ft/s
147 ft/s
Sneaky Example
Suppose you want to travel a distance of 2.0 miles at an average speed of 30. mph. You cover the first mile at a speed of 15. mph. What should your speed be in the second mile so you will average 30. miles per hour for the entire trip? (Hint: It’s faster than you think!)
Sneaky Example ContinuedS
peed
, mph
t, s
1.00 mile
15.
t1
Area = 1.00 miles = 15. t0or t0 = 1/15. miles = 4.0 minutes
Spe
ed, m
pht, s
2.00 mile
30.
t2
Area = 2.00 miles = 30. t0or t0 = 2.00/30. miles = 4.0 minutes
Sneaky Example MoralSo the second mile will have to be
covered in zero s! Or infinite speed!The lesson is you can’t average
averages! Only time and distance are preserved quantities!
Subway Example
A subway train is being planned using trains capable of 50.0 mph. How close can adjacent stations be so that the train will reach a speed of 50.0 mph given these characteristics:
acceleration = 6.0 ft/s2,
deceleration = - 4.0 ft/s2,
maximum speed is 50.0 mph,
and the train starts at V0 = 0.
Subway Example
Need: x1 and x2 corresponding to t1 and t2
Know: V0 (0) = 0 and V(t2) = 0; a1 = 6.0
ft/s2 and d2 = - 4.0 ft/s2, V1(t1) = V2(0) = 50.
mph = 73 ft/s
How: Speed/time graph
Subway ExampleS
peed
, ft/
s
6.0 ft/s2
-4.0 ft/s2
t1 t2t0= 0
73.
Subway Example
Solution: part 1:V1 = 73. ft/s
t1 = 73. /6.0 [ft/s] × [s2/ft] = 12.2 s
Area = x1 = ½ × 73 × 12.2 [ft/s][s]
= 445 ft
Similarly for t2 - t1 = -73./-4.0 [ft/s][s2s] = 18.3 s
Area = x2 = ½ × 73 × 18.3 [ft/s][s] = 668 ft
Station separation = 445 + 668 = 1,110 ft
Highway Capacity - Types of Traffic Flow
The first type is called uninterrupted flow, and is flow regulated by vehicle-vehicle interactions and interactions between vehicles and the roadway. For example, vehicles traveling on an interstate highway are participating in uninterrupted flow.
The second type of traffic flow is called interrupted flow. Interrupted flow is flow regulated by an external means, such as a traffic signal. Under interrupted flow conditions, vehicle-vehicle interactions and vehicle-roadway interactions play a secondary role in defining the traffic flow.
Traffic Flow Parameters Capacity (cars per hour). The number of cars that pass a
certain point during an hour.
Car speed, (miles per hour, mph). In our simple model we will assume all cars are traveling at the same speed.
Density (cars per mile). Suppose you took a snapshot of the highway from a helicopter and had previously marked two lines on the highway a mile apart. The number of cars you would count between these lines is the number of cars per mile.
You can easily write down the interrelationship among these variables by using dimensionally consistent units:
Capacity = Speed Density[cars/hour] = [miles/hour] [cars/mile]
Speed-Flow-Density Relationship
Suppose you are flying in a helicopter, take the snapshot mentioned above, and count that there are 160 cars per mile. Suppose further that your first partner determines that the cars are crawling along at only 2.5 miles per hour. Suppose your second partner is standing by the road with a watch counting cars as described earlier. For one lane of traffic, how many cars per hour will your second partner count?
Need: Capacity = ______ cars/hour.
Know–How: You already know a relationship between mph and cars per mile
Capacity = Speed × Density
[cars/hour] = [miles/hr] × [cars/mile]
Solve: Capacity = [2.5 miles/hr] × [160 cars/mile] = 400.cars/hour.
The “follow rule”Number of car lengths between cars
= [speed in mph]/[10 mph]
Use of rules Assume average length of a car is ~ 4.0 m
on. We could pack about 400. of these vehicles/mile bumper-to-bumper
• The second equation assumes the follow rule ofone spacing/10 mph
Effect of follow rule
Speed, mph
Density, cars/mile
Capacity, cars/hr
0 400 0 10 200 2000 20 133 2667 30 100 3000 40 80 3200 50 67 3333 60 57 3429 70 50 3500 80 44 3556 90 40 3600 100 36 3636
Follow rule and max cars on road
0
500
1000
1500
2000
2500
3000
3500
4000
0 25 50 75 100
Speed, mph
Cap
acit
y, c
ars/
hr
0
50
100
150
200
250
300
350
400
0 20 40 60 80 100
Speed, mphD
ensi
ty, c
ars/
mil
e
Summary Kinematics is study of relationships among speed,
distance, acceleration and time. For one-dimensional problems use a speed/time
graph since the slope of the line gives acceleration and the area under the curve gives the distance travelled.
For traffic analysis use:
An empirical rule + capacity relationship:
Capacity = Speed × Density [cars/hour] = [miles/hr] × [cars/mile]
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