(E) For this part we ignore the rst part of the motion
( : ) and consider what happens as the stone falls from
position , where it has zero vertical velocity, to position
. We dene the initial time as t B 0. Because the given
time for this part of the mot
y
Multiplying a Vector by a Scalar
If vector A is multiplied by a positive scalar quantity m, then the product mA is a vector
that has the same direction as A and magnitude mA. If vector A is multiplied by a negative scalar quantity m, then the product mA
Example 3.2 A Vacation Trip
A car travels 20.0 km due north and then 35.0 km in a direction 60.0 west of north, as shown in Figure 3.12a. Find
the magnitude and direction of the cars resultant
displacement.
Solution The vectors A and B drawn in Figure 3.1
y
We obtain the magnitude of R and the angle it makes with the x axis from its components, using the relationships
R R x2 R y2 (Ax Bx)2 (Ay By)2
Ry
Ay By
tan
Rx
Ax Bx
(3.16)
(3.17)
We can check this addition by components with a geometric construction, a
y
x
B
By
Bx
Unit Vectors
O
Figure 3.15 The component vectors of B in a coordinate system
that is tilted.
y
x
j
i
k
y
For example, consider a point lying in the xy plane and having Cartesian coordinates
(x, y), as in Figure 3.17. The point can be specied b
Example 3.3 The Sum of Two Vectors
Find the sum of two vectors A and B lying in the xy plane
and given by
A (2.0 2.0 m
i
j)
and
B (2.0 4.0 m
i
j)
Solution You may wish to draw the vectors to conceptualize
the situation. We categorize this as a simple plug
Bx B cos 60.0 (40.0 km)(0.500) 20.0 km
By B sin 60.0 (40.0 km)(0.866) 34.6 km
(B) Determine the components of the hikers resultant displacement R for the trip. Find an expression for R in terms
of unit vectors.
y(km)
0
Car
10
20
R y Ay By 17.7 km 34.6 km
In unitvector notation, R ( 95.3 232 km . Using
i
j)
Equations 3.16 and 3.17, we nd that the vector R has a
magnitude of 251 km and is directed 22.3 west of north.
To nalize the problem, note that the airplane can reach
city C from the starting point by r
6. A book is moved once around the perimeter of a tabletop
with the dimensions 1.0 m 2.0 m. If the book ends up
at its initial position, what is its displacement? What is the
distance traveled?
7. While traveling along a straight interstate highway you no
31. Consider the two vectors A 3 2 and B 4
i
j
i
j.
Calculate (a) A B, (b) A B, (c) A B, (d) A B,
and (e) the directions of A B and A B.
32. Consider the three displacement vectors A (3 3 m,
i
j)
B ( 4 m, and C ( 2i 5 m. Use the compoi
j)
j)
through a qua
45. As it passes over Grand Bahama Island, the eye of a hurricane 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 to
25.0 km/h. How far fr
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 oclock
on a Tuesday morning, and (b) at 5 oclock on a Saturday
morning. Explain your
perimeter of the rectangle. (b) Find the magnitude and
direction of the vector from the origin to the upper right
corner of the rectangle.
y
A
B
x
Figure P3.57
58. Find the sum of these four vector forces: 12.0 N to the
right at 35.0 above the horizontal,
In this chapter we explore the kinematics of a particle moving in two dimensions. Know-
ing the basics of two-dimensional motion will allow us to examinein future chaptersa
wide variety of motions, ranging from the motion of satellites in orbit to the mot
is actually encountered in physics and chemistry when an
array of electric charges (ions) exerts electric forces on an
atom at a central position in a molecule or in a crystal.
64. A rectangular parallelepiped has dimensions a, b, and c, as
in Figure P3.6
L
PITFALL PREVENTION
y
vi
While the vector addition discussed in Chapter 3 involves displacement vectors, vector addition
can be applied to any type of
vector quantity. Figure 4.4, for
example, shows the addition of
velocity vectors using the graphical ap
Mark C. Burnett/Photo Researchers, Inc.
y
Direction of v at
r1 r2 r3
"
'
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 quant
where x, y, and r change with time as the particle moves while the unit vectors and
i
j
remain constant. If the position vector is known, the velocity of the particle can be obtained from Equations 4.3 and 4.6, which give
dr
dx
dy
i
j vx vy
i
j
dt
dt
Vector Subtraction
B
A
B
C=AB
C=AB
B
A
(b)
(a)
Figure 3.11 (a) This construction shows how to subtract vector B from vector A. The
vector B is equal in magnitude to vector B and points in the opposite direction. To
subtract B from A, apply the rule of vec
A vector quantity is completely specied by a number and appropriate units plus a
direction.
Another example of a vector quantity is displacement, as you know from Chapter 2.
Suppose a particle moves from some point to some point along a straight path, as
Example 2.12
Interactive
Not a Bad Throw for a Rookie!
A stone thrown from the top of a building is given an initial
velocity of 20.0 m/s straight upward. The building is 50.0 m
high, and the stone just misses the edge of the roof on its
way down, as show
G E N E RA L P R O B L E M -S O LVI N G ST R AT E GY
Analyze
Conceptualize
The rst thing to do when approaching a problem is
to think about and understand the situation. Study carefully any diagrams, graphs, tables, or photographs
that accompany the probl
duration tn. From the denition of average velocity we see that the displacement
during any small interval, such as the one shaded in Figure 2.15, is given by
x n vxn t n where vxn is the average velocity in that interval. Therefore, the
displacement durin
vx
v x = a xt
a xtA
Figure 2.17 The velocitytime curve for a particle moving with a velocity that is proportional to
the time.
t
tA
acceleration), we nd that the displacement of the particle during the time interval
t 0 to t tA is equal to the area of the
QU ESTIONS
1. The speed of sound in air is 331 m/s. During the next
thunderstorm, try to estimate your distance from a lightning bolt by measuring the time lag between the ash and
the thunderclap. You can ignore the time it takes for the
light ash to reac
S U M MARY
Take a practice test for
this chapter by clicking the
Practice Test link at
http:/www.pse6.com.
After a particle moves along the x axis from some initial position xi to some nal position xf , its displacement is
x x f x i
(2.1)
The average velo
2. (a) Sand dunes in a desert move over time as sand is swept
up the windward side to settle in the lee side. Such walking dunes have been known to walk 20 feet in a year and
can travel as much as 100 feet per year in particularly
windy times. Calculate t
25. A particle moves along the x axis. Its position is given by the
equation x 2 3t 4t 2 with x in meters and t in seconds.
Determine (a) its position when it changes direction and
(b) its velocity when it returns to the position it had at t 0.
26. In the
13. Secretariat won the Kentucky Derby with times for successive quarter-mile segments of 25.2 s, 24.0 s, 23.8 s, and
23.0 s. (a) Find his average speed during each quarter-mile
segment. (b) Assuming that Secretariats instantaneous
speed at the nish line
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65. Setting a new world record in a 100-m race, Maggie and
Judy cross the nish line in a dead heat, both taking 10.2 s.
Accelerating uniformly, Maggie took 2.00 s and Judy 3.00 s
to attain maximum speed, which they main