© 2010 Pearson Education, Inc.
PowerPoint® Lectures for
College Physics: A Strategic Approach, Second Edition
Chapter 3
Vectors and
Motion in Two
Dimensions
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3 Vectors and Motion in Two Dimensions
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Reading Quiz
1. Ax is the __________ of the vector A.
A. magnitude
B. x-component
C. direction
D. size
E. displacement
Slide 3-7
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Answer
1. Ax is the __________ of the vector A.
A. magnitude
B. x-component
C. direction
D. size
E. displacement
Slide 3-8
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Reading Quiz
2. The acceleration vector of a particle in projectile motion
A. points along the path of the particle.
B. is directed horizontally.
C. vanishes at the particle’s highest point.
D. is directed down at all times.
E. is zero.
Slide 3-9
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Answer
Slide 3-10
2. The acceleration vector of a particle in projectile motion
A. points along the path of the particle.
B. is directed horizontally.
C. vanishes at the particle’s highest point.
D. is directed down at all times.
E. is zero.
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3. The acceleration vector of a particle in uniform circular motion
A. points tangent to the circle, in the direction of motion.
B. points tangent to the circle, opposite the direction of motion.
C. is zero.
D. points toward the center of the circle.
E. points outward from the center of the circle.
Reading Quiz
Slide 3-11
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Answer
Slide 3-12
3. The acceleration vector of a particle in uniform circular motion
A. points tangent to the circle, in the direction of motion.
B. points tangent to the circle, opposite the direction of motion.
C. is zero.
D. points toward the center of the circle.
E. points outward from the center of the circle.
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Vectors
Slide 3-13
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Note that a is
v
t
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Which of the vectors below best represents
the vector sum P + Q?
Checking Understanding
Slide 3-16
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Which of the vectors below best represents
the vector sum P + Q?
Answer
Slide 3-17
A.
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Checking Understanding
Which of the vectors below best represents
the difference P – Q?
Slide 3-18
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Which of the vectors below best represents
the difference P – Q?
Answer
Slide 3-19
B.
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Checking Understanding
Which of the vectors below best represents
the difference Q – P?
Slide 3-20
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Which of the vectors below best represents
the difference Q – P?
Answer
Slide 3-21
C.
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Component Vectors and Components
Slide 3-22
“component” = “projection”
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h
hosin
h
hacos
a
o
h
htan
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h
ho1sin
h
ha1cos
a
o
h
h1tan
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Component Vectors and Components
Slide 3-22
cos
sin
x
y
B B
B B
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Component Vectors and Components
Slide 3-22
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What are the x- and y-components of this vector?
A. 3, 2
B. 2, 3
C. 3, 2
D. 2, 3
E. 3, 2
Checking Understanding
Slide 3-23
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What are the x- and y-components of this vector?
Answer
Slide 3-24
A. 3, 2
B. 2, 3
C. 3, 2
D. 2, 3
E. 3, 2
Ways to represent a vector:
2, 3
3.6, 56
ˆ ˆ2 3
ˆ ˆ2 3
A
A
A x y
A i j
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What are the x- and y-components of this vector?
A. 3, 1
B. 3, 4
C. 3, 3
D. 4, 3
E. 3, 4
Checking Understanding
Slide 3-25
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What are the x- and y-components of this vector?
Answer
Slide 3-26
A. 3, 1
B. 3, 4
C. 3, 3
D. 4, 3
E. 3, 4
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The following vector has length 4.0 units.
What are the x- and y-components of this vector?
A. 3.5, 2.0
B. 2.0, 3.5
C. 3.5, 2.0
D. 2.0, 3.5
E. 3.5, 2.0
Checking Understanding
Slide 3-27
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The following vector has length 4.0 units.
What are the x- and y-components of this vector?
Answer
Slide 3-28
A. 3.5, 2.0
B. 2.0, 3.5
C. 3.5, 2.0
D. 2.0, 3.5
E. 3.5, 2.0
Solve by using common sense…
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The following vector has length 4.0 units.
What are the x- and y-components of this vector?
Answer
Slide 3-28
A. 3.5, 2.0
B. 2.0, 3.5
C. 3.5, 2.0
D. 2.0, 3.5
E. 3.5, 2.0
4 0 60 2 0
4 0 30 2 0
4 0 120 2 0
x
x
x
A
A
A
Solve by calculation:
. cos .
. sin .
. cos .
4 0 30 3 5
4 0 60 3 5
4 0 120 3 5
. cos .
. sin .
. sin .
y
y
y
A
A
A
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The following vector has length 4.0 units.
What are the x- and y-components of this vector?
A. 3.5, 2.0
B. 2.0, 3.5
C. 3.5, 2.0
D. 2.0, 3.5
E. 3.5, 2.0
Checking Understanding
Slide 3-29
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The following vector has length 4.0 units.
What are the x- and y-components of this vector?
Answer
Slide 3-30
A. 3.5, 2.0
B. 2.0, 3.5
C. 3.5, 2.0
D. 2.0, 3.5
E. 3.5, 2.0
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component perpendicular to the ramp?
Slide 3-31
30
cos30PP
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component perpendicular to the ramp?
Slide 3-31
30
sin30Q
Q
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component perpendicular to the ramp?
Slide 3-31
30
cos30R
R
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component perpendicular to the ramp?
Slide 3-31
30sin30S
S
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component parallel to the ramp?
Slide 3-31
30
sin30P
P
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component parallel to the ramp?
Slide 3-31
30
cos30Q
Q
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component parallel to the ramp?
Slide 3-31
30
sin30R
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Example Problem
The labeled vectors each have length 4 units. For each vector,
what is the component parallel to the ramp?
Slide 3-31
S
cos30S30
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Example Problems
The Manitou Incline was an extremely steep cog railway in the
Colorado mountains; cars climbed at a typical angle of 22 with
respect to the horizontal. What was the vertical elevation
change for the one-mile run along the track?
Slide 3-32
22
1 mih?oh
sin
(1 mi)sin(22 )
0.4 mi
oh h
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Example Problems
The maximum grade of interstate highways in the United
States is 6.0%, meaning a 6.0 meter rise for 100 m of
horizontal travel.
a. What is the angle with respect to the horizontal of the
maximum grade?
Slide 3-32
?
100 mah
6.0 moh 1 1 6.0 mtan tan 3.4
100 m
o
a
h
h
h
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Example Problems
The maximum grade of interstate highways in the United
States is 6.0%, meaning a 6.0 meter rise for 100 m of
horizontal travel.
b. Suppose a car is driving up a 6.0% grade on a mountain
road at 67 mph (30 m/s). How many seconds does it take
the car to increase its height by 100 m?
Slide 3-32
100 msin 1700 m
sin sin(6 )
oo
hh h h
1700 m
mi67
x xv t
t v
h
1609 m
1 mi
1 h
56 s
3600 s
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The diagram below shows two successive positions of a
particle; it’s a segment of a full motion diagram. Which of the
acceleration vectors best represents the acceleration between
vi and vf?
Checking Understanding
Slide 3-33
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The diagram below shows two successive positions of a
particle; it’s a segment of a full motion diagram. Which of the
acceleration vectors best represents the acceleration between
vi and vf?
Answer
Slide 3-34
D.
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A new ski area has opened that emphasizes the extreme nature
of the skiing possible on its slopes. Suppose an ad intones
“Free fall skydiving is the greatest rush you can
experience…but we’ll take you as close as you can get on land.
When you tip your skis down the slope of our steepest runs, you
can accelerate at up to 75% of the acceleration you’d
experience in free fall.” What angle slope could give such an
acceleration?
Example Problems: Motion on a Ramp
Slide 3-35
?
90
sin(90 ) a g
g
1 1
1
1
0.75
0.75 cos(90 )
cos(90 ) 0.75
cos cos(90 ) cos (0.75)
90 cos (0.75)
90 cos (0.75)
49
a g
g g
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Ski jumpers go down a long slope on slippery skis, achieving a
high speed before launching into air. The “in-run” is essentially a
ramp, which jumpers slide down to achieve the necessary
speed. A particular ski jump has a ramp length of 120 m tipped
at 21 with respect to the horizontal. What is the highest speed
that a jumper could reach at the bottom of such a ramp?
Example Problems: Motion on a Ramp
Slide 3-35
21
g
sing
120 m x
2 2
2
2
2
0
2
2
2(9.8 m/s )(sin 21 )(120 m)
29 m/s
xf xi x
xi
xf x
xf x
xf
xf
v v a x
v
v a x
v a x
v
v
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Example Problems: Relative Motion
An airplane pilot wants to fly due west from Spokane to
Seattle. Her plane moves through the air at 200 mph, but the
wind is blowing 40 mph due north. In what direction should
she point the plane—that is, in what direction should she fly
relative to the air?
Slide 3-36
200 mi/h40 mi/h
1 1 40 mi/htan tan
o
a
h
h 200 mi/h11 S of W
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Example Problems: Relative Motion
A skydiver jumps out of an airplane 1000 m directly above his
desired landing spot. He quickly reaches a steady speed,
falling through the air at 35 m/s. There is a breeze blowing at 7
m/s to the west. At what angle with respect to vertical does he
fall? When he lands, what will be his displacement from his
desired landing spot?
Slide 3-36
35 m/s
7 m/s
1 1 7 m/ssin sin
oh
h 35 m/s11.31 10
1000 m
x
opptan opp adj tan
adj
(1000 m)tan(11.31 )
200 m to the west
x
x
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Projectile Motion
The horizontal and vertical
components of the motion are
independent.
The horizontal motion is
constant; the vertical motion
is free fall:
Slide 3-37
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© 2010 Pearson Education, Inc. Slide 3-39
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Example Problem: Projectile Motion
In the movie Road Trip, some students are seeking to jump a car
across a gap in a bridge. One student, who professes to know
what he is talking about (“Of course I’m sure—with physics, I’m
always sure.”), says that they can easily make the jump.
Continued next slide…
Slide 3-40
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Example Problem: Projectile Motion
The car weighs 2100 pounds, with passengers and luggage.
Right before the gap, there’s a ramp that will launch the car at an
angle of 30°. The gap is 10 feet wide. He then suggests that they
should drive the car at a speed of 50 mph in order to make the
jump.
a. If the car actually went airborne at a speed of 50 mph at an
angle of 30° with respect to the horizontal, how far would it
travel before landing?
Slide 3-40
50 mi/hiv
cos iv
sin iv
30
21
2
0 0 0, so
10
2
0 or
2 2 sin10
2
f i yi y
f i
yi y
yi i
yi y
y
y y v t a t
y y y
t v a t
t
v vv a t t
a g
solution continued next slide...
© 2010 Pearson Education, Inc.
Example Problem: Projectile Motion
The car weighs 2100 pounds, with passengers and luggage.
Right before the gap, there’s a ramp that will launch the car at an
angle of 30°. The gap is 10 feet wide. He then suggests that they
should drive the car at a speed of 50 mph in order to make the
jump.
a. If the car actually went airborne at a speed of 50 mph at an
angle of 30° with respect to the horizontal, how far would it
travel before landing?
Slide 3-40
50 mi/hiv
cos iv
sin iv
30
2
2
1
22 2 sin
0, 0, and from the previous slide, , so
2 sin( cos )
2sin cos the "range equation"
xi xif
yi ixi
y
ixi if
if
x x v t a t
v vx a t
a g
vx v t v
g
vx
g
solution continued next slide...
© 2010 Pearson Education, Inc.
Example Problem: Projectile Motion
The car weighs 2100 pounds, with passengers and luggage.
Right before the gap, there’s a ramp that will launch the car at an
angle of 30°. The gap is 10 feet wide. He then suggests that they
should drive the car at a speed of 50 mph in order to make the
jump.
a. If the car actually went airborne at a speed of 50 mph at an
angle of 30° with respect to the horizontal, how far would it
travel before landing?
Slide 3-40
50 mi/hiv
cos iv
sin iv
30
mi50 iv
h
1609 m
1 mi
1 h
2 2
2
22 m/s3600 s
2 2(22 m/s)sin cos sin 30 cos30
9.8 m/s
100 cm 1 in 1 ft44 m 140 ft
1 m 2.54 cm 12 in
So the car could easily make the jump.
i
f
f
vx
g
x
© 2010 Pearson Education, Inc.
Example Problem: Projectile Motion
The car weighs 2100 pounds, with passengers and luggage.
Right before the gap, there’s a ramp that will launch the car at an
angle of 30°. The gap is 10 feet wide. He then suggests that they
should drive the car at a speed of 50 mph in order to make the
jump.
a. If the car actually went airborne at a speed of 50 mph at an
angle of 30° with respect to the horizontal, how far would it
travel before landing?
b. Does the mass of the car make any difference in your
calculation?
Slide 3-40
The mass does not make any difference, because the flight time depends only on g.
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Example Problem: Broad Jumps
A grasshopper can jump a distance of 30 in (0.76 m) from a
standing start. If the grasshopper takes off at the optimal angle
for maximum distance of the jump, what is the initial speed of
the jump?
Slide 3-41
2
2
45
Use the range equation:
2sin cos
2sin cos
9.8 m/s(0.76 m)
2sin(45 )cos(45 )
2.7 m/s
optimal
i
f i
i
i
v gx v x
g
v
v
cos iv
ivsin iv
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Example Problem: Broad Jumps
A grasshopper can jump a distance of 30 in (0.76 m) from a
standing start. If the grasshopper takes off at the optimal angle
for maximum distance of the jump, what is the initial speed of
the jump? Most animals jump at a lower angle than 45°.
Suppose the grasshopper takes off at 30° from the horizontal.
What jump speed is necessary to reach the noted distance?
Slide 3-41
2
30
9.8 m/s(0.76 m)
2sin cos 2sin(30 )cos(30 )
2.9 m/s
i
i
gv x
v
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Example Problem
Alan Shepard took a golf ball to the moon during one of the
Apollo missions, and used a makeshift club to hit the ball a
great distance. He described the shot as going for “miles and
miles.” A reasonable golf tee shot leaves the club at a speed of
64 m/s. Suppose you hit the ball at this speed at an angle of
30 with the horizontal in the moon’s gravitational acceleration
of 1.6 m/s2. How long is the ball in the air?
Slide 3-42
2
2 sin 2(64 m/s)sin(30 )
1.6 m/s
40 s
ivt
g
t
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Example Problem
Alan Shepard took a golf ball to the moon during one of the
Apollo missions, and used a makeshift club to hit the ball a
great distance. He described the shot as going for “miles and
miles.” A reasonable golf tee shot leaves the club at a speed of
64 m/s. Suppose you hit the ball at this speed at an angle of
30 with the horizontal in the moon’s gravitational acceleration
of 1.6 m/s2. How long is the ball in the air? How far would the
shot go?
Slide 3-42
2 2
2
2 2(64 m/s)sin cos sin(30 )cos(30 )
1.6 m/s
1 mi2200 m 1.4 miles!!!
1609 m
if
f
vx
g
x
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Circular Motion
There is an acceleration
because the velocity is
changing direction.
Slide 3-43
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Example Problems: Circular Motion
Two friends are comparing the acceleration of their vehicles.
Josh owns a Ford Mustang, which he clocks as doing 0 to 60
mph in a time of 5.6 seconds. Josie has a Mini Cooper that
she claims is capable of higher acceleration. When Josh
laughs at her, she proceeds to drive her car in a tight circle at
13 mph. Which car experiences a higher acceleration?
Slide 3-44
2
22
miniCooper
2
Mustang
mini Coopers have a turning radius of 17.5 ft = 5.3 m, so let's use 6 m:
mi 1 h 1609 m13
h 3600 s 1 mi5.6 m/s
6.0 m
mi 1609 m 1 h60
h 1 mi 3600 s 4.8 m/s5.6 s
va
r
va
t
The miniCooper wins!
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Example Problems: Circular Motion
Turning a corner at a typical large intersection is a city means
driving your car through a circular arc with a radius of about
25 m. If the maximum advisable acceleration of your vehicle
through a turn on wet pavement is 0.40 times the free-fall
acceleration, what is the maximum speed at which you should
drive through this turn?
Slide 3-44
2 2
22
2
0.40 0.40 9.8 m/s 3.9 m/s
m 1 mi 3600 s(3.9 m/s )(25 m) 9.9 22 mph
s 1609 m 1 h
a g
va v a r v a r
r
v
© 2010 Pearson Education, Inc. Slide 3-45
Summary
© 2010 Pearson Education, Inc. Slide 3-46
Summary