[ Pobierz całość w formacie PDF ]
It should also be emphasized that this model is
based on safe driving intervals between cars. If an
object were to drop from the back of a truck you are
following, you would need the safe distance to stop in
time to avoid hitting the object. On the other hand, if
the car in front of you, also traveling at 55 mph, has to
stop, and if both drivers have the same reaction time
and both cars decelerate at the same rate, then both
cars will need 231 ft to come to a stop. Hence, when
both cars come to a stop they will still be separated by
the distance of 231 ft. For this reason, in areas of very
Figure 4 The capacity of the road as a function of the
velocity of cars.
heavy traffic, many people do not leave the safe distance between them and the car in front. Instead, they get
closer and closer to the car in front of them until they are only separated by the reaction distance x . I call this the
R
kamikaze model, for obvious reasons. The kamikaze model allows more cars to travel at a greater velocity than are
allowed by the safe stopping distance model. The velocity of the cars as a function of the number of cars is found by
solving equation H3.2 with the v term, which is the term associated with the deceleration distance x set equal to
02 d
zero. The result is shown in figure 5, which compares the safe stopping distance model with the kamikaze model.
Notice that many more cars can now fit on the road. For example, in the safe stopping model, only 40 cars, each
traveling at 60 mph, can fit safely on this road. In the kamikaze model about 185 cars can fit on this road, but
certainly not safely. There will be only 44 ft between each car, and if you have a slower reaction time than that of
the car in front of you, you will almost certainly hit him when he steps on the brakes. This is the reason why there
are so many rear-end collisions on expressways. The number of cars on a real expressway falls somewhere
Chapter 3 Kinematics - The Study of Motion 3-35
between the extremes of these two models. Note that even in the kamikaze model, the velocity of the cars must
decrease with volume.
Figure 5 Comparison of traffic with the safe stopping Figure 6 Comparison of the capacity versus velocity
distance model and the kamikaze model. for the safe stopping distance model and
the kamikaze model.
The capacity of the expressway for the kamikaze model is found by setting the v term in equation H3.5 to
02
zero. The result is shown in figure 6. Notice that in the kamikaze model the capacity increases with velocity, and
there is no optimum speed for the maximum car flow. In practice, the actual capacity of an expressway lies
somewhere between these two extremes.
In conclusion, if your expressway is not much of an expressway, it is time to petition your legislators to
allocate more money for the widening of the expressway, or maybe it is time to move to a less populated part of the
country.
The Language of Physics
Kinematics Instantaneous velocity Acceleration due to gravity
The branch of mechanics that The velocity at a particular instant If air friction is ignored, all objects
describes the motion of a body of time. It is defined as the limit of that are dropped near the surface of
without regard to the cause of that the ratio of the change in the the earth, are accelerated toward
motion (p. ). displacement of the body to the the center of the earth with an
change in time, as the time interval acceleration of 9.80 m/s2.
Average velocity approaches zero. The magnitude of
The average rate at which the the instantaneous velocity is the Projectile motion
displacement vector changes with instantaneous speed of the moving The motion of a body thrown or
time. Since a displacement is a body (p. ). fired with an initial velocity v in a
vector, the velocity is also a vector gravitational field (p. ).
(p. ). Kinematic equations of linear
motion Trajectory
Average speed A set of equations that gives the The path through space followed by
The distance that a body moves per displacement and velocity of the a projectile (p. ).
unit time. Speed is a scalar moving body at
quantity (p. ). any instant of time, and the velocity Range of a projectile
of the moving body at any The horizontal distance from the
Constant velocity displacement, if the acceleration of point where the projectile is
A body moving in one direction in the body is a constant (p. ). launched to the point where it
such a way that it always travels returns to its launch height (p. ).
equal distances in equal times (p. ). Freely falling body
Any body that is moving under the
Acceleration influence of gravity only. Hence,
The rate at which the velocity of a any body that is dropped or thrown
moving body changes with time on the surface of the earth is a
(p. ). freely falling body (p. ).
3-36 Mechanics
Summary of Important Equations
Average velocity Velocity at any time y-displacement
v = "r = r - r (3.32) v = v + at (3.35) y = v t - 1 gt2 (3.39)
avg 2 1 0 0y
"t t - t 2
2 1
Displacement at any time
Acceleration r = v t + 1 at2 (3.34) x-component of velocity
a = "v = v - v (3.33) 2 v = v (3.40)
0 x 0x
"t t
Velocity at any displacement in the
y-component of velocity
x-direction
Instantaneous velocity in two or v = v - gt (3.41)
y 0y
more directions, which is a v2 = v + 2ax (3.16)
02
generalization of the instantaneous
y-component of velocity at any
velocity in one dimension Velocity at any displacement in the
height y
y-direction
"r v = v - 2gy (3.48)
y2 0y2
v = lim
v2 = v + 2ay (3.16)
02
"t’!0
"t
Range
(3.8)
For Projectile Motion
R = v sin 2¸ (3.47)
02
"x
v = lim x-displacement
g
"t’!0
"t
x = v t (3.38)
0x
Questions for Chapter 3
1. Discuss the difference 7. What effect would air person on the train and by a person
between distance and displacement. resistance have on the velocity of a on the station platform.
2. Discuss the difference body that is dropped near the 13. You are in free fall, and you
between speed and velocity. surface of the earth? let go of your watch. What is the
3. Discuss the difference 8. What is the acceleration of a relative velocity of the watch with
between average speed and projectile when its instantaneous respect to you?
instantaneous speed. vertical velocity is zero at the top of *14. What kind of motion is
*4. Although speed is the its trajectory? indicated by a graph of
magnitude of the instantaneous 9. Can an object have zero displacement versus time, if the
velocity, is the average speed equal velocity at the same time that it has slope of the curve is (a) horizontal,
to the magnitude of the average an acceleration? Explain and give (b) sloping upward to the right, and
velocity? some examples. (c) sloping downward?
5. Why can the kinematic 10. Can the velocity of an object *15. What kind of motion is
equations be used only for motion be in a different direction than the indicated by a graph of velocity
at constant acceleration? acceleration? Give some examples. versus time, if the slope of the curve
6. In dealing with average 11. Can you devise a means of is (a) horizontal, (b) sloping upward
velocities discuss the statement, using two clocks to measure your at a constant value, (c) sloping
Straight line motion at 60 km/hr reaction time? upward at a changing rate, [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl chiara76.opx.pl
It should also be emphasized that this model is
based on safe driving intervals between cars. If an
object were to drop from the back of a truck you are
following, you would need the safe distance to stop in
time to avoid hitting the object. On the other hand, if
the car in front of you, also traveling at 55 mph, has to
stop, and if both drivers have the same reaction time
and both cars decelerate at the same rate, then both
cars will need 231 ft to come to a stop. Hence, when
both cars come to a stop they will still be separated by
the distance of 231 ft. For this reason, in areas of very
Figure 4 The capacity of the road as a function of the
velocity of cars.
heavy traffic, many people do not leave the safe distance between them and the car in front. Instead, they get
closer and closer to the car in front of them until they are only separated by the reaction distance x . I call this the
R
kamikaze model, for obvious reasons. The kamikaze model allows more cars to travel at a greater velocity than are
allowed by the safe stopping distance model. The velocity of the cars as a function of the number of cars is found by
solving equation H3.2 with the v term, which is the term associated with the deceleration distance x set equal to
02 d
zero. The result is shown in figure 5, which compares the safe stopping distance model with the kamikaze model.
Notice that many more cars can now fit on the road. For example, in the safe stopping model, only 40 cars, each
traveling at 60 mph, can fit safely on this road. In the kamikaze model about 185 cars can fit on this road, but
certainly not safely. There will be only 44 ft between each car, and if you have a slower reaction time than that of
the car in front of you, you will almost certainly hit him when he steps on the brakes. This is the reason why there
are so many rear-end collisions on expressways. The number of cars on a real expressway falls somewhere
Chapter 3 Kinematics - The Study of Motion 3-35
between the extremes of these two models. Note that even in the kamikaze model, the velocity of the cars must
decrease with volume.
Figure 5 Comparison of traffic with the safe stopping Figure 6 Comparison of the capacity versus velocity
distance model and the kamikaze model. for the safe stopping distance model and
the kamikaze model.
The capacity of the expressway for the kamikaze model is found by setting the v term in equation H3.5 to
02
zero. The result is shown in figure 6. Notice that in the kamikaze model the capacity increases with velocity, and
there is no optimum speed for the maximum car flow. In practice, the actual capacity of an expressway lies
somewhere between these two extremes.
In conclusion, if your expressway is not much of an expressway, it is time to petition your legislators to
allocate more money for the widening of the expressway, or maybe it is time to move to a less populated part of the
country.
The Language of Physics
Kinematics Instantaneous velocity Acceleration due to gravity
The branch of mechanics that The velocity at a particular instant If air friction is ignored, all objects
describes the motion of a body of time. It is defined as the limit of that are dropped near the surface of
without regard to the cause of that the ratio of the change in the the earth, are accelerated toward
motion (p. ). displacement of the body to the the center of the earth with an
change in time, as the time interval acceleration of 9.80 m/s2.
Average velocity approaches zero. The magnitude of
The average rate at which the the instantaneous velocity is the Projectile motion
displacement vector changes with instantaneous speed of the moving The motion of a body thrown or
time. Since a displacement is a body (p. ). fired with an initial velocity v in a
vector, the velocity is also a vector gravitational field (p. ).
(p. ). Kinematic equations of linear
motion Trajectory
Average speed A set of equations that gives the The path through space followed by
The distance that a body moves per displacement and velocity of the a projectile (p. ).
unit time. Speed is a scalar moving body at
quantity (p. ). any instant of time, and the velocity Range of a projectile
of the moving body at any The horizontal distance from the
Constant velocity displacement, if the acceleration of point where the projectile is
A body moving in one direction in the body is a constant (p. ). launched to the point where it
such a way that it always travels returns to its launch height (p. ).
equal distances in equal times (p. ). Freely falling body
Any body that is moving under the
Acceleration influence of gravity only. Hence,
The rate at which the velocity of a any body that is dropped or thrown
moving body changes with time on the surface of the earth is a
(p. ). freely falling body (p. ).
3-36 Mechanics
Summary of Important Equations
Average velocity Velocity at any time y-displacement
v = "r = r - r (3.32) v = v + at (3.35) y = v t - 1 gt2 (3.39)
avg 2 1 0 0y
"t t - t 2
2 1
Displacement at any time
Acceleration r = v t + 1 at2 (3.34) x-component of velocity
a = "v = v - v (3.33) 2 v = v (3.40)
0 x 0x
"t t
Velocity at any displacement in the
y-component of velocity
x-direction
Instantaneous velocity in two or v = v - gt (3.41)
y 0y
more directions, which is a v2 = v + 2ax (3.16)
02
generalization of the instantaneous
y-component of velocity at any
velocity in one dimension Velocity at any displacement in the
height y
y-direction
"r v = v - 2gy (3.48)
y2 0y2
v = lim
v2 = v + 2ay (3.16)
02
"t’!0
"t
Range
(3.8)
For Projectile Motion
R = v sin 2¸ (3.47)
02
"x
v = lim x-displacement
g
"t’!0
"t
x = v t (3.38)
0x
Questions for Chapter 3
1. Discuss the difference 7. What effect would air person on the train and by a person
between distance and displacement. resistance have on the velocity of a on the station platform.
2. Discuss the difference body that is dropped near the 13. You are in free fall, and you
between speed and velocity. surface of the earth? let go of your watch. What is the
3. Discuss the difference 8. What is the acceleration of a relative velocity of the watch with
between average speed and projectile when its instantaneous respect to you?
instantaneous speed. vertical velocity is zero at the top of *14. What kind of motion is
*4. Although speed is the its trajectory? indicated by a graph of
magnitude of the instantaneous 9. Can an object have zero displacement versus time, if the
velocity, is the average speed equal velocity at the same time that it has slope of the curve is (a) horizontal,
to the magnitude of the average an acceleration? Explain and give (b) sloping upward to the right, and
velocity? some examples. (c) sloping downward?
5. Why can the kinematic 10. Can the velocity of an object *15. What kind of motion is
equations be used only for motion be in a different direction than the indicated by a graph of velocity
at constant acceleration? acceleration? Give some examples. versus time, if the slope of the curve
6. In dealing with average 11. Can you devise a means of is (a) horizontal, (b) sloping upward
velocities discuss the statement, using two clocks to measure your at a constant value, (c) sloping
Straight line motion at 60 km/hr reaction time? upward at a changing rate, [ Pobierz całość w formacie PDF ]