© Arndt/Premium Stock/PictureQuest 4.1 The Position, Velocity, and Acceleration Vectors 4.2 Two-Dimensional Motion with Constant Acceleration 4.3 Projectile Motion 4.4 The Particle in Un
Trang 164 Chapter 3 Vectors
If a vector has an x component A x and a y
compo-nent A y, the vector can be expressed in unit–vector
vector pointing in the positive x direction and is a
unit vector pointing in the positive y direction.
Because and are iˆ jˆ unit vectors, 0iˆ0 0jˆ0 1
jˆ iˆ A
S
A xiˆ A yjˆ
A
S
We can find the resultant of two or more vectors by
resolving all vectors into their x and y components, adding their resultant x and y components, and then
using the Pythagorean theorem to find the magnitude
of the resultant vector We can find the angle that the
resultant vector makes with respect to the x axis by
using a suitable trigonometric function
Summary
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D E F I N I T I O N S
Scalar quantities are those that have only a numerical value and no associated direction Vector quantities have
both magnitude and direction and obey the laws of vector addition The magnitude of a vector is always a positive
number
CO N C E P T S A N D P R I N C I P L E S
When two or more vectors are added together, they
must all have the same units and all of them must be
the same type of quantity We can add two vectors
and graphically In this method (Active Fig 3.6), the
resultant vector runs from the tail of to
denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question
1 OYes or no: Is each of the following quantities a vector?
(a) force (b) temperature (c) the volume of water in a
can (d) the ratings of a TV show (e) the height of a
building (f) the velocity of a sports car (g) the age of
the Universe
2. A book is moved once around the perimeter of a tabletop
with dimensions 1.0 m 2.0 m If the book ends up at its
initial position, what is its displacement? What is the
dis-tance traveled?
3 OFigure Q3.3 shows two vectors, and Which of the
possibilities (a) through (d) is the vector , or (e) is
in Figure Q3.4 Rank these situations according to the magnitude of the total displacement of the tool, putting the situation with the greatest resultant magnitude first If the total displacement is the same size in two situations, give those letters equal ranks.
5 OLet represent a velocity vector pointing from the
ori-gin into the second quadrant (a) Is its x component tive, negative, or zero? (b) Is its y component positive,
posi-negative, or zero? Let BS represent a velocity vector
point-A
S
Trang 2ing from the origin into the fourth quadrant (c) Is its x
component positive, negative, or zero? (d) Is its y
compo-nent positive, negative, or zero? (e) Consider the vector
What, if anything, can you conclude about
quad-rants it must be in or cannot be in? (f) Now consider the
vector What, if anything, can you conclude about
quadrants it must be in or cannot be in?
6 O (i) What is the magnitude of the vector
? (a) 0 (b) 10 m/s (c) 10 m/s (d) 10 (e) 10 (f) 14.1 m/s (g) undefined (ii) What
is the y component of this vector? (Choose from among
the same answers.)
7 O A submarine dives from the water surface at an angle
of 30° below the horizontal, following a straight path 50
m long How far is the submarine then below the water
surface? (a) 50 m (b) sin 30° (c) cos 30° (d) tan 30°
(e) (50 m)/sin 30° (f) (50 m)/cos 30° (g) (50 m)/
tan 30° (h) (50 m)sin 30° (i) (50 m)cos 30°
(j) (50 m)tan 30° (k) (sin 30°)/50 m (l) (cos 30°)/50 m
(m) (tan 30°)/50 m (n) 30 m (o) 0 (p) none of these
9 OVector lies in the xy plane (i) Both of its components
will be negative if it lies in which quadrant(s)? Choose all that apply (a) the first quadrant (b) the second quad- rant (c) the third quadrant (d) the fourth quadrant
(ii)For what orientation(s) will its components have site signs? Choose from among the same possibilities.
oppo-10. If the component of vector along the direction of vector
is zero, what can you conclude about the two vectors?
11. Can the magnitude of a vector have a negative value? Explain.
12. Is it possible to add a vector quantity to a scalar quantity? Explain.
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with additional quizzing and conceptual questions.
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Guide ; denotes coached solution with hints available at www.thomsonedu.com; denotes developing symbolic reasoning;
denotes asking for qualitative reasoning; denotes computer useful in solving problem
Section 3.1 Coordinate Systems
1. The polar coordinates of a point are r 5.50 m and
u 240° What are the Cartesian coordinates of this point?
2. Two points in a plane have polar coordinates (2.50 m,
30.0°) and (3.80 m, 120.0°) Determine (a) the Cartesian
coordinates of these points and (b) the distance between
them.
3. A fly lands on one wall of a room The lower left-hand
cor-ner of the wall is selected as the origin of a two-dimensional
Cartesian coordinate system If the fly is located at the
point having coordinates (2.00, 1.00) m, (a) how far is it
from the corner of the room? (b) What is its location in
polar coordinates?
4. The rectangular coordinates of a point are given by (2, y),
and its polar coordinates are (r, 30°) Determine y and r.
5. Let the polar coordinates of the point (x, y) be (r, u).
Determine the polar coordinates for the points (a) (x, y),
(b) (2x, 2y), and (c) (3x, 3y).
Section 3.2 Vector and Scalar Quantities Section 3.3 Some Properties of Vectors
6. A plane flies from base camp to lake A, 280 km away in the direction 20.0° north of east After dropping off sup- plies it flies to lake B, which is 190 km at 30.0° west of north from lake A Graphically determine the distance and direction from lake B to the base camp.
7. A surveyor measures the distance across a straight river by the following method: starting directly across from a tree
on the opposite bank, she walks 100 m along the bank to establish a baseline Then she sights across to the tree The angle from her baseline to the tree is 35.0° How wide is the river?
river-8. A force of magnitude 6.00 units acts on an object at
the origin in a direction 30.0° above the positive x axis A
second force of magnitude 5.00 units acts on the object
in the direction of the positive y axis Graphically find the
magnitude and direction of the resultant force F .
2 = intermediate; 3 = challenging; = SSM/SG; = ThomsonNOW; = symbolic reasoning; = qualitative reasoning
(j)6 cm (k) none of these answers (ii) What is the y
component of this vector? (Choose from among the same answers.)
Trang 39. A skater glides along a circular path of radius 5.00 m If
he coasts around one half of the circle, find (a) the
mag-nitude of the displacement vector and (b) how far he
skated (c) What is the magnitude of the displacement if
he skates all the way around the circle?
10. Arbitrarily define the “instantaneous vector height” of a
person as the displacement vector from the point halfway
between his or her feet to the top of the head Make an
order-of-magnitude estimate of the total vector height of
all the people in a city of population 100 000 (a) at
10 o’clock on a Tuesday morning and (b) at 5 o’clock on
a Saturday morning Explain your reasoning.
11. Each of the displacement vectors and shown in
Figure P3.11 has a magnitude of 3.00 m Graphically find
(a) , (b) , (c) , and (d) Report
all angles counterclockwise from the positive x axis.
18. A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east (a) What is her resultant displacement? (b) What is the total distance she travels?
19. Obtain expressions in component form for the position vectors having the following polar coordinates: (a) 12.8 m, 150° (b) 3.30 cm, 60.0° (c) 22.0 in., 215°
20. A displacement vector lying in the xy plane has a
magni-tude of 50.0 m and is directed at an angle of 120° to the
positive x axis What are the rectangular components of
this vector?
21. While exploring a cave, a spelunker starts at the entrance and moves the following distances She goes 75.0 m north, 250 m east, 125 m at an angle 30.0° north of east, and 150 m south Find her resultant displacement from the cave entrance.
22. A map suggests that Atlanta is 730 miles in a direction of 5.00° north of east from Dallas The same map shows that Chicago is 560 miles in a direction of 21.0° west of north from Atlanta Modeling the Earth as flat, use this informa- tion to find the displacement from Dallas to Chicago.
23. A man pushing a mop across a floor causes it to undergo two displacements The first has a magnitude of 150 cm
and makes an angle of 120° with the positive x axis The
resultant displacement has a magnitude of 140 cm and is
directed at an angle of 35.0° to the positive x axis Find
the magnitude and direction of the second displacement.
24. Given the vectors
, (a) draw the vector sum and the vector difference (b) Calculate and , first in terms of unit vectors and then in terms of polar coordi- nates, with angles measured with respect to the x axis.
25. Consider the two vectors and Calculate (a) , (b) , (c) , (d) , and (e) the directions of and
26. A snow-covered ski slope makes an angle of 35.0° with the horizontal When a ski jumper plummets onto the hill, a parcel of splashed snow projects to a maximum position
of 5.00 m at 20.0° from the vertical in the uphill direction
as shown in Figure P3.26 Find the components of its maximum position (a) parallel to the surface and (b) per- pendicular to the surface.
12. Three displacements are due south,
due west, and east of north.
Construct a separate diagram for each of the following
possible ways of adding these vectors: ;
; Explain what you can conclude from comparing the diagrams.
13. A roller-coaster car moves 200 ft horizontally and then
rises 135 ft at an angle of 30.0° above the horizontal It
next travels 135 ft at an angle of 40.0° downward What is
its displacement from its starting point? Use graphical
techniques.
14. A shopper pushing a cart through a store moves 40.0 m
down one aisle, then makes a 90.0° turn and moves 15.0 m.
He then makes another 90.0° turn and moves 20.0 m.
(a) How far is the shopper away from his original
posi-tion? (b) What angle does his total displacement make
with his original direction? Notice that we have not
speci-fied whether the shopper turned right or left Explain
how many answers are possible for parts (a) and (b) and
give the possible answers.
Section 3.4 Components of a Vector and Unit Vectors
15. A vector has an x component of 25.0 units and a y
component of 40.0 units Find the magnitude and
direc-tion of this vector.
16. A person walks 25.0° north of east for 3.10 km How far
would she have to walk due north and due east to arrive
at the same location?
17. A minivan travels straight north in the right lane of a
divided highway at 28.0 m/s A camper passes the
mini-van and then changes from the left into the right lane As
it does so, the camper’s path on the road is a straight
dis-placement at 8.50° east of north To avoid cutting off the
minivan, the north–south distance between the camper’s
2 = intermediate; 3 = challenging; = SSM/SG; = ThomsonNOW; = symbolic reasoning; = qualitative reasoning
27. A particle undergoes the following consecutive ments: 3.50 m south, 8.20 m northeast, and 15.0 m west What is the resultant displacement?
Trang 4displace-28. In a game of American football, a quarterback takes the
ball from the line of scrimmage, runs backward a distance
of 10.0 yards, and then runs sideways parallel to the line
of scrimmage for 15.0 yards At this point, he throws a
for-ward pass 50.0 yards straight downfield perpendicular to
the line of scrimmage What is the magnitude of the
foot-ball’s resultant displacement?
29. A novice golfer on the green takes three strokes to sink
the ball The successive displacements of the ball are 4.00 m
to the north, 2.00 m northeast, and 1.00 m at 30.0° west
of south Starting at the same initial point, an expert
golfer could make the hole in what single displacement?
30. Vector has x and y components of 8.70 cm and 15.0 cm,
respectively; vector has x and y components of 13.2 cm
and 6.60 cm, respectively If , what are
the components of ?
31. The helicopter view in Figure P3.31 shows two people
pulling on a stubborn mule Find (a) the single force that
is equivalent to the two forces shown and (b) the force
that a third person would have to exert on the mule to
make the resultant force equal to zero The forces are
measured in units of newtons (symbolized N).
37. The vector has x, y, and z components of 8.00, 12.0, and
–4.00 units, respectively (a) Write a vector expression for
in unit–vector notation (b) Obtain a unit–vector sion for a vector one-fourth the length of pointing in the same direction as (c) Obtain a unit–vector expres- sion for a vector three times the length of pointing in the direction opposite the direction of
expres-38. You are standing on the ground at the origin of a nate system An airplane flies over you with constant
coordi-velocity parallel to the x axis and at a fixed height of 7.60
10 3m At time t 0 the airplane is directly above you so that the vector leading from you to it is
At t 30.0 s the position vector ing from you to the airplane is
lead- Determine the magnitude and
orienta-tion of the airplane’s posiorienta-tion vector at t 45.0 s.
39. A radar station locates a sinking ship at range 17.3 km and bearing 136° clockwise from north From the same station, a rescue plane is at horizontal range 19.6 km, 153° clockwise from north, with elevation 2.20 km (a) Write the position vector for the ship relative to the plane, letting represent east, north, and up (b) How far apart are the plane and ship?
40. (a) Vector has magnitude 17.0 cm and is directed 27.0° counterclockwise from the x axis Express it in unit–vector
notation (b) Vector has magnitude 17.0 cm and is directed 27.0° counterclockwise from the y axis Express
it in unit–vector notation (c) Vector has magnitude 17.0 cm and is directed 27.0° clockwise from the y axis.
Express it in unit–vector notation.
41. Vector has a negative x component 3.00 units in length and a positive y component 2.00 units in length.
(a) Determine an expression for in unit–vector tion (b) Determine the magnitude and direction of (c) What vector when added to gives a resultant vec-
nota-tor with no x component and a negative y component
4.00 units in length?
42. As it passes over Grand Bahama Island, the eye of a cane is moving in a direction 60.0° north of west with a speed of 41.0 km/h Three hours later the course of the hurricane suddenly shifts due north, and its speed slows
hurri-to 25.0 km/h How far from Grand Bahama is the eye 4.50 h after it passes over the island?
43. Three displacement vectors of a croquet ball are shown
in Figure P3.43, where , , and 0C Find (a) the resultant in unit–vector
Figure P3.31
Figure P3.36
32. Use the component method to add the vectors and
shown in Figure P3.11 Express the resultant in
unit–vector notation.
33. Vector has x, y, and z components of 4.00, 6.00, and
3.00 units, respectively Calculate the magnitude of and
the angles makes with the coordinate axes.
34. Consider the three displacement vectors ,
, and Use the ponent method to determine (a) the magnitude and
com-direction of the vector and (b) the
mag-nitude and direction of
35. Given the displacement vectors
and , find the magnitudes of the
vectors (a) and (b) , also
express-ing each in terms of its rectangular components.
36. In an assembly operation illustrated in Figure P3.36, a
robot moves an object first straight upward and then also
to the east, around an arc forming one quarter of a circle
of radius 4.80 cm that lies in an east–west vertical plane.
The robot then moves the object upward and to the
north, through one-quarter of a circle of radius 3.70 cm
that lies in a north–south vertical plane Find (a) the
Trang 5notation and (b) the magnitude and direction of the
resultant displacement.
68 Chapter 3 Vectors
map of the successive displacements (b) What total tance did she travel? (c) Compute the magnitude and direction of her total displacement The logical structure
dis-of this problem and dis-of several problems in later chapters was suggested by Alan Van Heuvelen and David Maloney,
American Journal of Physics 67(3) 252–256, March 1999.
47. Two vectors and have precisely equal magnitudes For the magnitude of to be 100 times larger than the magnitude of , what must be the angle between them?
48. Two vectors and have precisely equal magnitudes For the magnitude of to be larger than the magnitude
of by the factor n, what must be the angle between
them?
49. An air-traffic controller observes two aircraft on his radar screen The first is at altitude 800 m, horizontal distance 19.2 km, and 25.0° south of west The second is at altitude
1 100 m, horizontal distance 17.6 km, and 20.0° south of west What is the distance between the two aircraft? (Place
the x axis west, the y axis south, and the z axis vertical.)
50.The biggest stuffed animal in the world is a snake 420 m long, constructed by Norwegian children Suppose the snake is laid out in a park as shown in Figure P3.50, form- ing two straight sides of a 105° angle, with one side 240 m long Olaf and Inge run a race they invent Inge runs directly from the tail of the snake to its head, and Olaf starts from the same place at the same moment but runs along the snake If both children run steadily at 12.0 km/h, Inge reaches the head of the snake how much earlier than Olaf?
y
Figure P3.45 Point A is a fraction f of the distance from the
ini-tial point (5, 3) to the final point (16, 12).
and b such that (b) A student has
learned that a single equation cannot be solved to
deter-mine values for more than one unknown in it How would
you explain to him that both a and b can be determined
from the single equation used in part (a)?
45. Are we there yet? In Figure P3.45, the line segment
repre-sents a path from the point with position vector
to the point with location
Point A is along this path, a fraction f of the way to the
destination (a) Find the position vector of point A in
terms of f (b) Evaluate the expression from part (a) in
the case f 0 Explain whether the result is reasonable.
(c) Evaluate the expression for f 1 Explain whether the
46. On December 1, 1955, Rosa Parks (1913–2005), an icon of
the early civil rights movement, stayed seated in her bus
seat when a white man demanded it Police in
Mont-gomery, Alabama, arrested her On December 5, blacks
began refusing to use all city buses Under the leadership
of the Montgomery Improvement Association, an efficient
system of alternative transportation sprang up
immedi-ately, providing blacks with approximately 35 000 essential
trips per day through volunteers, private taxis, carpooling,
and ride sharing The buses remained empty until they
were integrated under court order on December 21, 1956.
In picking up her riders, suppose a driver in downtown
Montgomery traverses four successive displacements
repre-sented by the expression
Here b represents one city block, a convenient unit of
dis-tance of uniform size; is east; and is north (a) Draw aiˆ jˆ
13.00b cos 50°2 iˆ 13.00b sin 50°2 jˆ 15.00b2 jˆ
16.30b2 iˆ 14.00b cos 40°2 iˆ 14.00b sin 40°2 jˆ
51. A ferryboat transports tourists among three islands It sails from the first island to the second island, 4.76 km away, in
a direction 37.0° north of east It then sails from the ond island to the third island in a direction 69.0° west of north Finally, it returns to the first island, sailing in a direction 28.0° east of south Calculate the distance between (a) the second and third islands and (b) the first and third islands.
sec-52. A vector is given by Find (a) the
magni-tudes of the x, y, and z components, (b) the magnitude of
, and (c) the angles between and the x, y, and z axes.
53. A jet airliner, moving initially at 300 mi/h to the east, suddenly enters a region where the wind is blowing at
100 mi/h toward the direction 30.0° north of east What are the new speed and direction of the aircraft relative to the ground?
54. Let measured from the horizontal Let at some angle u (a) Find the magnitude
of as a function of u (b) From the answer to part (a), for what value of u does 0A take on its maximum
Trang 6value? What is this maximum value? (c) From the answer
to part (a), for what value of u does take on its
minimum value? What is this minimum value? (d)
With-out reference to the answer to part (a), argue that the
answers to each of parts (b) and (c) do or do not make
sense.
55. After a ball rolls off the edge of a horizontal table at time
t 0, its velocity as a function of time is given by
The ball’s displacement away from the edge of the table,
during the time interval of 0.380 s during which it is in
flight, is given by
To perform the integral, you can use the calculus theorem
You can think of the units and unit vectors as constants,
represented by A and B Do the integration to calculate
the displacement of the ball.
56 Find the sum of these four vector forces: 12.0 N to the
right at 35.0° above the horizontal, 31.0 N to the left at
55.0° above the horizontal, 8.40 N to the left at 35.0°
below the horizontal, and 24.0 N to the right at 55.0°
below the horizontal Follow these steps Guided by a
sketch of this situation, explain how you can simplify the
calculations by making a particular choice for the
direc-tions of the x and y axes What is your choice? Then add
the vectors by the component method.
57. A person going for a walk follows the path shown in
Fig-ure P3.57 The total trip consists of four straight-line
paths At the end of the walk, what is the person’s
resul-tant displacement measured from the starting point?
1A Bf 1x2 2 dx A dx B f 1x2 dx
¢ Sr
0.380 s 0
to some origin as shown in Figure P3.59 His ship’s log
instructs you to start at tree A and move toward tree B, but to cover only one-half the distance between A and B Then move toward tree C, covering one-third the distance between your current location and C Next move toward tree D, covering one-fourth the distance between where you are and D Finally, move toward tree E, covering one-fifth the distance between you and E, stop, and dig (a) Assume
you have correctly determined the order in which the
pirate labeled the trees as A, B, C, D, and E as shown in
the figure What are the coordinates of the point where
his treasure is buried? (b) What If? What if you do not
really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order
of the trees, for instance to B(30 m, 20 m), A(60 m,
80 m), E(10 m, –10 m), C(40 m, 30 m), and
D(70 m, 60 m)? State reasoning to show the answer does not depend on the order in which the trees are labeled.
End
x y
200 m 60.0 30.0150 m
300 m
100 m Start
Figure P3.57
E
y
x A
B
C
D
Figure P3.59
58. The instantaneous position of an object is specified by
its position vector leading from a fixed origin to the
location of the object, modeled as a particle Suppose for
a certain object the position vector is a function of time,
given by , where r is in meters and t is
in seconds Evaluate What does it represent about
the object?
59. Long John Silver, a pirate, has buried his treasure on
an island with five trees, located at the points (30.0 m,
60. Consider a game in which N children position
them-selves at equal distances around the circumference of a circle At the center of the circle is a rubber tire Each child holds a rope attached to the tire and, at a signal, pulls on his or her rope All children exert forces of the
same magnitude F In the case N 2, it is easy to see that the net force on the tire will be zero because the two oppositely directed force vectors add to zero Similarly, if
N 4, 6, or any even integer, the resultant force on the tire must be zero because the forces exerted by each pair
of oppositely positioned children will cancel When an odd number of children are around the circle, it is not as obvious whether the total force on the central tire will be zero (a) Calculate the net force on the tire in the case
N 3 by adding the components of the three force
vec-tors Choose the x axis to lie along one of the ropes.
(b) What If? State reasoning that will determine the net
force for the general case where N is any integer, odd or
even, greater than one Proceed as follows Assume the total force is not zero Then it must point in some partic- ular direction Let every child move one position clock- wise Give a reason that the total force must then have a
direction turned clockwise by 360°/N Argue that the
total force must nevertheless be the same as before Explain what the contradiction proves about the magnitude
of the force This problem illustrates a widely useful nique of proving a result “by symmetry,” by using a bit
tech-of the mathematics tech-of group theory The particular
situa-tion is actually encountered in physics and chemistry
2 = intermediate; 3 = challenging; = SSM/SG; = ThomsonNOW; = symbolic reasoning; = qualitative reasoning
Trang 7when an array of electric charges (ions) exerts electric
forces on an atom at a central position in a molecule or
in a crystal.
61. Vectors and have equal magnitudes of 5.00 The sum
of and is the vector Determine the angle
between and
62. A rectangular parallelepiped has dimensions a, b, and c as
shown in Figure P3.62 (a) Obtain a vector expression for
the face diagonal vector What is the magnitude of this
vector? (b) Obtain a vector expression for the body
diago-nal vector Notice that , , and make a right
tri-angle Prove that the magnitude of R is 1 a2 b2 c2
Answers to Quick Quizzes
3.1 Scalars: (a), (d), (e) None of these quantities has a
direc-tion Vectors: (b), (c) For these quantities, the direction
is necessary to specify the quantity completely.
3.2 (c) The resultant has its maximum magnitude A B
12 8 20 units when vector is oriented in the same
direction as vector The resultant vector has its
mini-mum magnitude A B 12 8 4 units when vector
is oriented in the direction opposite vector
3.3 (b) and (c) To add to zero, the vectors must point in
opposite directions and have the same magnitude.
in which case the magnitude of the vector is equal to the absolute value of that component.
3.5 (c) The magnitude of is 5 units, the same as the z
com-ponent Answer (b) is not correct because the magnitude
of any vector is always a positive number, whereas the y
Trang 8Lava spews from a volcanic eruption Notice the parabolic paths of embers
projected into the air All projectiles follow a parabolic path in the absence
of air resistance (© Arndt/Premium Stock/PictureQuest)
4.1 The Position, Velocity, and Acceleration Vectors
4.2 Two-Dimensional Motion with Constant Acceleration
4.3 Projectile Motion
4.4 The Particle in Uniform Circular Motion
4.5 Tangential and Radial Acceleration
4.6 Relative Velocity and Relative Acceleration
In this chapter, we explore the kinematics of a particle moving in two dimensions.
Knowing the basics of two-dimensional motion will allow us—in future chapters—
to examine a variety of motions ranging from the motion of satellites in orbit to
the motion of electrons in a uniform electric field We begin by studying in greater
detail the vector nature of position, velocity, and acceleration We then treat
pro-jectile motion and uniform circular motion as special cases of motion in two
dimensions We also discuss the concept of relative motion, which shows why
observers in different frames of reference may measure different positions and
velocities for a given particle
Acceleration Vectors
In Chapter 2, we found that the motion of a particle along a straight line is
com-pletely known if its position is known as a function of time Let us now extend this
idea to two-dimensional motion of a particle in the xy plane We begin by describing
the position of the particle by its position vector , drawn from the origin of some
coordinate system to the location of the particle in the xy plane, as in Figure 4.1
(page 72) At time t i, the particle is at point , described by position vector At
some later time t f, it is at point , described by position vector The path fromSr
Trang 9to is not necessarily a straight line As the particle moves from to in thetime interval t t f t i, its position vector changes from to As we learned inChapter 2, displacement is a vector, and the displacement of the particle is the dif-
ference between its final position and its initial position We now define the placement vector for a particle such as the one in Figure 4.1 as being the dif-ference between its final position vector and its initial position vector:
We define the average velocity of a particle during the time interval t as
the displacement of the particle divided by the time interval:
(4.2)
Multiplying or dividing a vector quantity by a positive scalar quantity such as t
changes only the magnitude of the vector, not its direction Because displacement
is a vector quantity and the time interval is a positive scalar quantity, we concludethat the average velocity is a vector quantity directed along
The average velocity between points is independent of the path taken That is
because average velocity is proportional to displacement, which depends only onthe initial and final position vectors and not on the path taken As with one-dimensional motion, we conclude that if a particle starts its motion at some pointand returns to this point via any path, its average velocity is zero for this tripbecause its displacement is zero Consider again our basketball players on the court
in Figure 2.2 (page 21) We previously considered only their one-dimensionalmotion back and forth between the baskets In reality, however, they move over atwo-dimensional surface, running back and forth between the baskets as well asleft and right across the width of the court Starting from one basket, a givenplayer may follow a very complicated two-dimensional path Upon returning to theoriginal basket, however, a player’s average velocity is zero because the player’s dis-placement for the whole trip is zero
Consider again the motion of a particle between two points in the xy plane as
shown in Figure 4.2 As the time interval over which we observe the motion
Figure 4.1 A particle moving in the
xy plane is located with the position
vector drawn from the origin
to the particle The displacement of
the particle as it moves from to
in the time interval t t f t iis
equal to the vector ¢rS
to , the respective displacements and ding time intervals become smaller and smaller In the limit that the end point approaches , t approaches zero
correspon-and the direction of approaches that of the line gent to the curve at By definition, the instantaneous velocity at is directed along this tangent line.
Trang 10becomes smaller and smaller—that is, as is moved to and then to , and
so on—the direction of the displacement approaches that of the line tangent to
the path at The instantaneous velocity is defined as the limit of the average
velocity as t approaches zero:
(4.3)
That is, the instantaneous velocity equals the derivative of the position vector with
respect to time The direction of the instantaneous velocity vector at any point in a
particle’s path is along a line tangent to the path at that point and in the direction
of motion
The magnitude of the instantaneous velocity vector of a particle is called
the speed of the particle, which is a scalar quantity.
As a particle moves from one point to another along some path, its
instanta-neous velocity vector changes from at time t ito at time t f Knowing the velocity
at these points allows us to determine the average acceleration of the particle The
average acceleration of a particle is defined as the change in its instantaneous
velocity vector divided by the time interval t during which that change occurs:
(4.4)
Because is the ratio of a vector quantity and a positive scalar quantity t, we
conclude that average acceleration is a vector quantity directed along As
indi-cated in Figure 4.3, the direction of is found by adding the vector (the
negative of ) to the vector because, by definition,
When the average acceleration of a particle changes during different time
inter-vals, it is useful to define its instantaneous acceleration The instantaneous
acceler-ation is defined as the limiting value of the ratio as t approaches zero:
(4.5)
In other words, the instantaneous acceleration equals the derivative of the velocity
vector with respect to time
Various changes can occur when a particle accelerates First, the magnitude of the
velocity vector (the speed) may change with time as in straight-line (one-dimensional)
motion Second, the direction of the velocity vector may change with time even if
its magnitude (speed) remains constant as in two-dimensional motion along a
curved path Finally, both the magnitude and the direction of the velocity vector
may change simultaneously.
Although the vector addition
dis-cussed in Chapter 3 involves
displace-ment vectors, vector addition can be
applied to any type of vector
quan-tity Figure 4.3, for example, shows
the addition of velocity vectors using
the graphical approach.
x y
Figure 4.3 A particle moves from position to position Its velocity vector changes from to
The vector diagrams at the upper right show two ways of determining the vector from the
ini-tial and final velocities.
Trang 11Quick Quiz 4.1 Consider the following controls in an automobile: gas pedal,brake, steering wheel What are the controls in this list that cause an acceleration
of the car? (a) all three controls (b) the gas pedal and the brake (c) only thebrake (d) only the gas pedal
Constant Acceleration
In Section 2.5, we investigated one-dimensional motion of a particle under stant acceleration Let us now consider two-dimensional motion during which theacceleration of a particle remains constant in both magnitude and direction As weshall see, this approach is useful for analyzing some common types of motion.Before embarking on this investigation, we need to emphasize an importantpoint regarding two-dimensional motion Imagine an air hockey puck moving in astraight line along a perfectly level, friction-free surface of an air hockey table Fig-ure 4.4a shows a motion diagram from an overhead point of view of this puck.Recall that in Section 2.4 we related the acceleration of an object to a force on theobject Because there are no forces on the puck in the horizontal plane, it moves
con-with constant velocity in the x direction Now suppose you blow a puff of air on the puck as it passes your position, with the force from your puff of air exactly in the y direction Because the force from this puff of air has no component in the x direction, it causes no acceleration in the x direction It only causes a momentary acceleration in the y direction, causing the puck to have a constant y component
of velocity once the force from the puff of air is removed After your puff of air on
the puck, its velocity component in the x direction is unchanged, as shown in
Fig-ure 4.4b The generalization of this simple experiment is that motion in two
dimensions can be modeled as two independent motions in each of the two dicular directions associated with the x and y axes That is, any influence in the y direction does not affect the motion in the x direction and vice versa.
perpen-The position vector for a particle moving in the xy plane can be written
(4.6)
where x, y, and change with time as the particle moves while the unit vectors
and remain constant If the position vector is known, the velocity of the particlecan be obtained from Equations 4.3 and 4.6, which give
(4.7) v
of the figure, which demonstrates that motion in two dimensions can be modeled as two independent motions in perpendicular directions.
Trang 12Because the acceleration of the particle is assumed constant in this discussion,
its components a x and a yalso are constants Therefore, we can model the particle
as a particle under constant acceleration independently in each of the two
direc-tions and apply the equadirec-tions of kinematics separately to the x and y components of
the velocity vector Substituting, from Equation 2.13, v xf v xi a x t and v yf v yi
a y t into Equation 4.7 to determine the final velocity at any time t, we obtain
(4.8)
This result states that the velocity of a particle at some time t equals the vector sum
of its initial velocity at time t 0 and the additional velocity acquired at time
t as a result of constant acceleration Equation 4.8 is the vector version of Equation
2.13
Similarly, from Equation 2.16 we know that the x and y coordinates of a particle
moving with constant acceleration are
Substituting these expressions into Equation 4.6 (and labeling the final position
vector ) gives
(4.9)
which is the vector version of Equation 2.16 Equation 4.9 tells us that the position
vector of a particle is the vector sum of the original position , a displacement
arising from the initial velocity of the particle and a displacement resulting
from the constant acceleration of the particle
Graphical representations of Equations 4.8 and 4.9 are shown in Active Figure
4.5 The components of the position and velocity vectors are also illustrated in the
figure Notice from Active Figure 4.5a that is generally not along the direction
of either or because the relationship between these quantities is a vector
expression For the same reason, from Active Figure 4.5b we see that is generally
not along the direction of or Finally, notice that and are generally not in
the same direction
v yi t
ri
at2 1
a x t2
1
x i
ACTIVE FIGURE 4.5
Vector representations and components of (a) the velocity and (b) the position of a particle moving
with a constant acceleration
Sign in at www.thomsonedu.comand go to ThomsonNOW to investigate the effect of different initial
positions and velocities on the final position and velocity (for constant acceleration).
a
S
Trang 13Substitute numerical values:
Finalize Notice that the x component of velocity increases in time while the y component remains constant; this
result is consistent with what we predicted
(B)Calculate the velocity and speed of the particle at t 5.0 s
SOLUTION
Analyze
Evaluate the result from Equation (1) at t 5.0 s:
Determine the angle u that makes with the x axis at
Finalize The negative sign for the angle u indicates that the velocity vector is directed at an angle of 21° below the
positive x axis Notice that if we calculate v i from the x and y components of , we find that v f v i Is that consistentwith our prediction?
(C)Determine the x and y coordinates of the particle at any time t and its position vector at this time.
A particle starts from the origin at t 0 with an initial velocity having an x component of 20 m/s and a y component of
15 m/s The particle moves in the xy plane with an x component of acceleration only, given by a x 4.0 m/s2
(A)Determine the total velocity vector at any time
SOLUTION
Conceptualize The components of the initial velocity tell
us that the particle starts by moving toward the right and
downward The x component of velocity starts at 20 m/s
and increases by 4.0 m/s every second The y
compo-nent of velocity never changes from its initial value of
15 m/s We sketch a motion diagram of the situation
in Figure 4.6 Because the particle is accelerating in the
x direction, its velocity component in this direction
increases and the path curves as shown in the diagram
Notice that the spacing between successive images
increases as time goes on because the speed is increasing The placement of the acceleration and velocity vectors inFigure 4.6 helps us further conceptualize the situation
Categorize Because the initial velocity has components in both the x and y directions, we categorize this problem as one involving a particle moving in two dimensions Because the particle only has an x component of acceleration, we model
it as a particle under constant acceleration in the x direction and a particle under constant velocity in the y direction.
Analyze To begin the mathematical analysis, we set v xi 20 m/s, v yi 15 m/s, a x 4.0 m/s2, and a y 0
x y
Figure 4.6 (Example 4.1) Motion diagram for the particle.
Trang 144.3 Projectile Motion
Anyone who has observed a baseball in motion has observed projectile motion
The ball moves in a curved path and returns to the ground Projectile motion of
an object is simple to analyze if we make two assumptions: (1) the free-fall
acceler-ation is constant over the range of motion and is directed downward,1and (2) the
effect of air resistance is negligible.2With these assumptions, we find that the path
of a projectile, which we call its trajectory, is always a parabola as shown in Active
Figure 4.7 We use these assumptions throughout this chapter.
The expression for the position vector of the projectile as a function of time
fol-lows directly from Equation 4.9, with :
(4.10) r
Section 4.3 Projectile Motion 77
Express the position vector of the particle at any time t: Sr
f x fiˆ y fjˆ 3 120t 2.0t22 iˆ 15t jˆ4 m
Finalize Let us now consider a limiting case for very large values of t.
1 This assumption is reasonable as long as the range of motion is small compared with the radius of the
Earth (6.4 10 6 m) In effect, this assumption is equivalent to assuming that the Earth is flat over the
range of motion considered.
2This assumption is generally not justified, especially at high velocities In addition, any spin imparted
to a projectile, such as that applied when a pitcher throws a curve ball, can give rise to some very
inter-esting effects associated with aerodynamic forces, which will be discussed in Chapter 14.
The parabolic path of a projectile that leaves the origin with a velocity The velocity vector changes
with time in both magnitude and direction This change is the result of acceleration in the negative y
direction The x component of velocity remains constant in time because there is no acceleration along
the horizontal direction The y component of velocity is zero at the peak of the path.
Sign in at www.thomsonedu.comand go to ThomsonNOW to change launch angle and initial speed.
You can also observe the changing components of velocity along the trajectory of the projectile.
Acceleration at the Highest Point
As discussed in Pitfall Prevention 2.8, many people claim that the acceleration of a projectile at the topmost point of its trajectory is zero This mistake arises from con- fusion between zero vertical veloc- ity and zero acceleration If the projectile were to experience zero acceleration at the highest point, its velocity at that point would not change; rather, the projectile would move horizontally at con- stant speed from then on! That does not happen, however, because
the acceleration is not zero
any-where along the trajectory.
What If? What if we wait a very long time and then
observe the motion of the particle? How would we describe
the motion of the particle for large values of the time?
Answer Looking at Figure 4.6, we see the path of the
particle curving toward the x axis There is no reason to
assume that this tendency will change, which suggests
that the path will become more and more parallel to the
x axis as time grows large Mathematically, Equation (1)
shows that the y component of the velocity remains stant while the x component grows linearly with t There- fore, when t is very large, the x component of the veloc- ity will be much larger than the y component, suggesting
con-that the velocity vector becomes more and more parallel
to the x axis Both x f and y fcontinue to grow with time,
although x fgrows much faster
Trang 15where the initial x and y components of the velocity of the projectile are
displace-In Section 4.2, we stated that two-dimensional motion with constant acceleration
can be analyzed as a combination of two independent motions in the x and y tions, with accelerations a x and a y Projectile motion can also be handled in this
direc-way, with zero acceleration in the x direction and a constant acceleration in the y direction, a y g Therefore, when analyzing projectile motion, model it to be the
superposition of two motions: (1) motion of a particle under constant velocity in the horizontal direction and (2) motion of a particle under constant acceleration (free fall) in the vertical direction The horizontal and vertical components of a
projectile’s motion are completely independent of each other and can be handled
separately, with time t as the common variable for both components.
Quick Quiz 4.2 (i)As a projectile thrown upward moves in its parabolic path(such as in Fig 4.8), at what point along its path are the velocity and accelerationvectors for the projectile perpendicular to each other? (a) nowhere (b) the high-est point (c) the launch point (ii)From the same choices, at what point are thevelocity and acceleration vectors for the projectile parallel to each other?
Horizontal Range and Maximum Height of a Projectile
Let us assume a projectile is launched from the origin at t i 0 with a positive v yi
component as shown in Figure 4.9 and returns to the same horizontal level Twopoints are especially interesting to analyze: the peak point , which has Cartesian
coordinates (R/2, h), and the point , which has coordinates (R, 0) The distance
R is called the horizontal range of the projectile, and the distance h is its maximum height Let us find h and R mathematically in terms of v i, ui , and g.
We can determine h by noting that at the peak v y 0 Therefore, we can use
the y component of Equation 4.8 to determine the time tat which the projectilereaches the peak:
Substituting this expression for t into the y component of Equation 4.9 and replacing y y with h, we obtain an expression for h in terms of the magnitude
and direction of the initial velocity vector:
78 Chapter 4 Motion in Two Dimensions
A welder cuts holes through a heavy
metal construction beam with a hot
torch The sparks generated in the
process follow parabolic paths.
Figure 4.8 The position vector of
a projectile launched from the origin
whose initial velocity at the origin is
The vector would be the
dis-placement of the projectile if gravity
were absent, and the vector is its
vertical displacement from a
straight-line path due to its downward
Figure 4.9 A projectile launched
over a flat surface from the origin at
t i 0 with an initial velocity The
maximum height of the projectile is
h, and the horizontal range is R At
, the peak of the trajectory, the
par-ticle has coordinates (R/2, h).
v
S
i