# Halliday Resnick Walker 7ª edição, resolução dos exercícios em ingles

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```If vv is the vertical
component of the velocity and \u3b8 is the angle between the direction of motion and the
vertical, then tan \u3b8 = vh/vv. Thus vv = vh/tan \u3b8 = (30 m/s)/tan 70° = 10.9 m/s. The speed of
a raindrop is v v vh v= + = + =2 2 30 10 9 32( ( . m / s) m / s) m / s2 2 .
64. We make use of Eq. 4-44 and Eq. 4-45.
The velocity of Jeep P relative to A at the instant is (in m/s)
\u2c6 \u2c6 \u2c6 \u2c640.0(cos60 i sin 60 j) 20.0i 34.6 j.PAv = ° + ° = +&
Similarly, the velocity of Jeep B relative to A at the instant is (in m/s)
\u2c6 \u2c6 \u2c6 \u2c620.0(cos30 i sin 30 j) 17.3i 10.0 j.BAv = ° + ° = +&
Thus, the velocity of P relative to B is (in m/s)
\u2c6 \u2c6 \u2c6 \u2c6 \u2c6 \u2c6(20.0i 34.6 j) (17.3i 10.0 j) 2.68i+24.6j.PB PA BAv v v= \u2212 = + \u2212 + =& & &
(a) The magnitude of PBv
&
is 2 2| | (2.68) (24.6) 24.8 m/s.PBv = + =&
(b) The direction of PBv
&
is 1tan (24.6 / 2.68) 83.8\u3b8 \u2212= = ° north of east (or 6.2º east of
north).
(c) The acceleration of P is \u2c6 \u2c6 \u2c6 \u2c60.400(cos 60.0 i sin 60.0 j) 0.200i 0.346j,PAa = ° + ° = +&
and PA PBa a=
& &
. Thus, we have 2| | 0.400 m/s .PBa =&
(d) The direction is 60.0° north of east (or 30.0° east of north).
65. Here, the subscript W refers to the water. Our coordinates are chosen with +x being
east and +y being north. In these terms, the angle specifying east would be 0° and the
angle specifying south would be \u201390° or 270°. Where the length unit is not displayed, km
is to be understood.
(a) We have & & &v v vA W A B B W = + , so that
&
vA B = (22 \u2220 \u2013 90°) \u2013 (40 \u2220 37°) = (56 \u2220 \u2013 125°)
in the magnitude-angle notation (conveniently done with a vector-capable calculator in
polar mode). Converting to rectangular components, we obtain
\u2c6 \u2c6( 32km/h) i (46 km/h) j .A Bv = \u2212 \u2212&
Of course, this could have been done in unit-vector notation from the outset.
(b) Since the velocity-components are constant, integrating them to obtain the position is
straightforward ( )& & &r r v dt\u2212 = z0
\u2c6 \u2c6(2.5 32 ) i (4.0 46 ) jr t t= \u2212 + \u2212&
with lengths in kilometers and time in hours.
(c) The magnitude of this &r is r t t= \u2212 + \u2212( . ) ( . )2 5 32 4 0 462 2 . We minimize this by
taking a derivative and requiring it to equal zero \u2014 which leaves us with an equation for t
dr
dt
t
t t
=
\u2212
\u2212 + \u2212
=
1
2
6286 528
2 5 32 4 0 46
0
2 2( . ) ( . )
which yields t = 0.084 h.
(d) Plugging this value of t back into the expression for the distance between the ships (r),
we obtain r = 0.2 km. Of course, the calculator offers more digits (r = 0.225\u2026), but they
are not significant; in fact, the uncertainties implicit in the given data, here, should make
the ship captains worry.
66. We construct a right triangle starting from the clearing on the south bank, drawing a
line (200 m long) due north (upward in our sketch) across the river, and then a line due
west (upstream, leftward in our sketch) along the north bank for a distance
(82 m) (1.1 m/s)t+ , where the t-dependent contribution is the distance that the river will
carry the boat downstream during time t.
The hypotenuse of this right triangle (the arrow in our sketch) also depends on t and on
the boat\u2019s speed (relative to the water), and we set it equal to the Pythagorean \u201csum\u201d of
the triangle\u2019s sides:
4 0 200 82 112 2. .b g b gt t= + +
46724 180 4 14 8 02+ \u2212 =. . .t t
We solve this and find a positive value: t = 62.6 s. The angle between the northward
(200 m) leg of the triangle and the hypotenuse (which is measured \u201cwest of north\u201d) is
then given by
\u3b8 = +FHG
I
KJ =
F
HG
I
KJ = °
\u2212 \u2212tan
.
tan .1 1
82 11
200
151
200
37t
67. Using displacement = velocity x time (for each constant-velocity part of the trip),
along with the fact that 1 hour = 60 minutes, we have the following vector addition
exercise (using notation appropriate to many vector capable calculators):
(1667 m \u2220 0º) + (1333 m \u2220 \u221290º) + (333 m \u2220 180º) + (833 m \u2220 \u221290º) + (667 m \u2220 180º)
+ (417 m \u2220 \u221290º) = (2668 m \u2220 \u221276º).
(a) Thus, the magnitude of the net displacement is 2.7 km.
(b) Its direction is 76° clockwise (relative to the initial direction of motion).
68. We compute the coordinate pairs (x, y) from x = v0 cos\u3b8t and x v t gt= \u22120 12 2sin \u3b8 for t
= 20 s and the speeds and angles given in the problem.
(a) We obtain (in kilometers)
x y x y
x y x y
A A B B
C C D D
, . , . , . , .
, . , . , . , .
b g b g b g b g
b g b g b g b g
= =
= =
101 0 56 121 151
14 3 2 68 16 4 399
and (xE, yE) = (18.5, 5.53) which we plot in the next part.
(b) The vertical (y) and horizontal (x) axes are in kilometers. The graph does not start at
the origin. The curve to \u201cfit\u201d the data is not shown, but is easily imagined (forming the
\u201ccurtain of death\u201d).
69. Since 22 0 2y yv v g y= \u2212 \u2206 , and vy=0 at the target, we obtain
( )( )0 2 9.80 5.00 9.90 m/syv = =
(a) Since v0 sin \u3b80 = v0y, with v0 = 12.0 m/s, we find \u3b80 = 55.6°.
(b) Now, vy = v0y \u2013 gt gives t = 9.90/9.80 = 1.01 s. Thus, \u2206x = (v0 cos \u3b80)t = 6.85 m.
(c) The velocity at the target has only the vx component, which is equal to v0x = v0 cos \u3b80
= 6.78 m/s.
70. Let vo = 2\u3c0(0.200)/.00500 \u2248 251 m/s (using Eq. 4-35) be the speed it had in circular
motion and \u3b8o = (1 hr)(360º/12 hr [for full rotation]) = 30.0º. Then Eq. 4-25 leads to
y = (2.50 )tan30.0º \u2013 (9.8)(2.50)
2
2(251cos(30º))2 \u2248 1.44 m
which means its height above the floor is (1.44 + 1.20) m = 2.64 m.
71. The (x,y) coordinates (in meters) of the points are A = (15, \u221215), B = (30, \u221245), C =
(20, \u221215), and D = (45, 45). The respective times are tA = 0, tB = 300 s, tC = 600 s, and tD
= 900 s. Average velocity is defined by Eq. 4-8. Each displacement \u2206r\u2192 is understood to
originate at point A.
(a) The average velocity having the least magnitude (5.0/600) is for the displacement
ending at point C: | | 0.0083 m/s.
avgv =
&
(b) The direction of
avgv
&
is 0°(measured counterclockwise from the +x axis).
(c) The average velocity having the greatest magnitude ( 152 + 302300 ) is for the
displacement ending at point B: | | 0.11 m/s.
avgv =
&
(d) The direction of
avgv
&
is 297° (counterclockwise from +x) or \u221263° (which is
equivalent to measuring 63° clockwise from the +x axis).
72. From the figure, the three displacements can be written as (in unit of meters)
1 1 1 1
2 2 1 2 1 2
3 3 3 2 1 3 2 1
\u2c6 \u2c6 \u2c6 \u2c6 \u2c6 \u2c6(cos i sin j) 5.00(cos30 i sin 30 j) 4.33i 2.50j
\u2c6 \u2c6 \u2c6 \u2c6[cos(180 )i sin(180 )j] 8.00(cos160 i sin160 j)
\u2c6 \u2c67.52i 2.74j
\u2c6 \u2c6[cos(360 )i sin(360 )j] 12.0(
d d
d d
d d
\u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8
\u3b8 \u3b8 \u3b8 \u3b8 \u3b8 \u3b8
= + = ° + ° = +
= ° + \u2212 + ° + \u2212 = ° + °
= \u2212 +
= ° \u2212 \u2212 + + ° \u2212 \u2212 + =
&
&
&
\u2c6 \u2c6cos 260 i sin 260 j)
\u2c6 \u2c62.08i 11.8j
° + °
= \u2212 \u2212
where the angles are measured from the +x axis. The net displacement is
1 2 3
\u2c6 \u2c65.27i 6.58j.d d d d= + + = \u2212 \u2212& & & &
(a) The magnitude of the net displacement is 2 2| | ( 5.27) ( 6.58) 8.43 m.d = \u2212 + \u2212 =&
(b) The direction of d
&
is
1 1 6.58tan tan 51.3 or 231 .
5.27
y
x
d
d
\u3b8 \u2212 \u2212§ · \u2212§ ·= = = ° °¨ ¸ ¨ ¸