Chapter 3
Vectors & 2-Dimensional
Motion
Chapter 3 Vectors & 2-Dimensional Motion
1
3.1 Vectors & Scalars
Revisited
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Vector: magnitude & direction
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Displacement
Velocity
Acceleration
Scalar: magnitude but no direction
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Temperature
Speed
Time intervals
Chapter 3 Vectors & 2-Dimensional Motion
2
3.2 Vector Properties
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Vector Format
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Handwritten: A
Printed: A, bold font
Scalar Format: A, italics
Chapter 3 Vectors & 2-Dimensional Motion
3
3.2 Vector Properties
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Vector Equality
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A & B are equal if they have the same magnitude
& direction.
Equal vectors can be moved parallel to itself
without being affected
Chapter 3 Vectors & 2-Dimensional Motion
4
3.2 Vector Properties

Which of these vectors have the same
MAGNITUDE?
Chapter 3 Vectors & 2-Dimensional Motion
5
3.2 Vector Properties

Which of these vectors have the same DIRECTION?
Chapter 3 Vectors & 2-Dimensional Motion
6
3.2 Vector Properties
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Adding Vectors
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Must have same units
Graphical Methods
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Triangular method of addition
Parallelogram method of addition
Sum is independent of order of addition
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
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A+B=B+A
Commutative law of addition
Component Method
Chapter 3 Vectors & 2-Dimensional Motion
7
3.2 Vector Properties
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Triangle Method
Chapter 3 Vectors & 2-Dimensional Motion
8
3.2 Vector Properties
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Parallelogram Method
Chapter 3 Vectors & 2-Dimensional Motion
9
3.2 Vector Properties
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Negative of a Vector
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Subtracting Vectors
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Same magnitude  opposite direction
A + (-A) = 0
A – B = A + (-B)
Multiplying/Dividing by a scalar

4A, A/5
Chapter 3 Vectors & 2-Dimensional Motion
10
3.2 Vector Properties
•
•
•
Adding 2 Vectors
Adding 3 Vectors
Subtracting Vectors
Chapter 3 Vectors & 2-Dimensional Motion
11
3.3 Vector Components
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V = Vx + V y
Vx = V cos Ө
Vy = V sin Ө
Vy
Vx
Chapter 3 Vectors & 2-Dimensional Motion
12
3.3 Vector Components

http://id.mind.net/~zona/mstm/physics/mecha
nics/vectors/components/vectorComponents.
html
Chapter 3 Vectors & 2-Dimensional Motion
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Vector Tutorial
Khan Academy Vector Tutorial
Aircraft Takeoff Problem
Chapter 3 Vectors & 2-Dimensional Motion
14
Practice Problems

Find the x and y components of the following
vectors:
240 N at 330º
34 m/s at 210º
15 m at 12º
20 m/s2 at 90º
Chapter 3 Vectors & 2-Dimensional Motion
15
Practice Problems

Find the x and y components of the following
vectors:
240 N at 330º Fy = 207.85 Fx = 120
34 m/s at 210º
15 m at 12º
20 m/s2 at 90º
Chapter 3 Vectors & 2-Dimensional Motion
16
Practice Problems

Find the x and y components of the following
vectors:
240 N at 330º Fy = 207.85 Fx = 120
34 m/s at 210º Vy = 17.0
Vx = 29.44
15 m at 12º
20 m/s2 at 90º
Chapter 3 Vectors & 2-Dimensional Motion
17
Practice Problems

Find the x and y components of the following
vectors:
240 N at 330º Fy = 207.85 Fx = 120
34 m/s at 210º Vy = 17.0
Vx = 29.44
15 m at 12º
xy = 3.12
xx = 1.4
20 m/s2 at 90º
Chapter 3 Vectors & 2-Dimensional Motion
18
Practice Problems

Find the x and y components of the following
vectors:
240 N at 330º Fy = -120
Fx = 207.85
34 m/s at 210º Vy = 17.0
Vx = 29.44
15 m at 12º
xy = 3.12
xx = 1.4
20 m/s2 at 90º
ay = 20.0
ax = 0
Chapter 3 Vectors & 2-Dimensional Motion
19
Component Method
• Adding vectors using “trig” & “arithmetic”
Step 1: Find all x and y components
Step 2: Add up all the x components
Add up all the y components
Step 3: Using the “new” x and y components
find the “new” resulting vector!
Step 4: Sanity check
Vector & Projectile Motion
Practice Problems

Find the resultant of the following 2 vectors:
1) 100 units due west and 2) 200 units 30o
north of east.
21
Vector & Projectile Motion
Practice Problems

Find the resultant of the following 2 vectors:
1) 100 units due east and 2) 200 units 30o
north of east.

124 units 54o north of west
22
Vector & Projectile Motion
Practice Problems

An ant on a picnic table travels 30 cm
eastward, then 25 cm northward and finally
15 cm westward. What is its directional
displacement with respect to its original
position?
23
Vector & Projectile Motion
Practice Problems

An ant on a picnic table travels 30 cm
eastward, then 25 cm northward and finally
15 cm westward. What is its directional
displacement with respect to its original
position?

59o north of east
24
Vector & Projectile Motion
Practice Problems

A boy pulls a sled across a level field by
exerting a force of 110 newtons at an angle
of 30o with the ground. What are the parallel
and perpendicular components, respectively,
of this force with respect to the ground?
25
Vector & Projectile Motion
Practice Problems

A boy pulls a sled across a level field by
exerting a force of 110 newtons at an angle
of 30o with the ground. What are the parallel
and perpendicular components, respectively,
of this force with respect to the ground?

95 newtons, 55 newtons
26
Vector & Projectile Motion
Practice Problems

I walk 6 miles in a straight line in a direction
north of east and I end up 2 miles east and
several miles north. How many degrees north
of east have I walked?
27
Vector & Projectile Motion
Practice Problems

I walk 6 miles in a straight line in a direction
north of east and I end up 2 miles east and
several miles north. How many degrees north
of east have I walked?
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71o
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Practice Problems

From the x and y components given, find
the direction and magnitude of the resultant.
Fy = 120 N, Fx = 345 N
vy = 31 m/s, vx = 8 m/s
Chapter 3 Vectors & 2-Dimensional Motion
29
Practice Problems


A soccer ball is kicked with a horizontal
velocity of 11.3 m/s and a vertical velocity of
3.5 m/s. What is the magnitude and direction
of the ball's velocity?
A shot putter applies a force of 415 N to a
shot at an angle of 37º. What are the
horizontal and vertical components of this
force?
Chapter 3 Vectors & 2-Dimensional Motion
30
Homework
Page(s) 76 & 77
#1,2,5,7,10,13,15,18,19
Due Tomorrow whether you have class or not!
Projectile Motion
Chapter 3 Vectors & 2-Dimensional Motion
32
Chapter 3
Projectile Motion
Chapter 3 Vectors & 2-Dimensional Motion
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Chapter 3
Projectile Motion

Animated Projectile Motion
Chapter 3 Vectors & 2-Dimensional Motion
34
3.5 Projectile Motion
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Can be described as a superposition of two
independent motions in the x and y directions
If air resistance is negligible, horizontal component
remains constant because there is no acceleration
in the horizontal direction.
Vertical component is equal to the free-fall
acceleration, g.
Vertical component of velocity and y-direction
displacement are identical to a freely falling object.
Chapter 3 Vectors & 2-Dimensional Motion
35
3.5 Projectile Motion

If you are carrying a ball and running at
constant speed and wish to throw the ball so
that you can catch it as it comes back down,
should you (a) throw the ball at a 45o angle
above the horizontal and maintain the same
speed, (b) throw the ball straight up in the air
and slow down to catch it, or (c) throw the
ball straight up in the air and maintain the
same speed?
Chapter 3 Vectors & 2-Dimensional Motion
36
3.5 Projectile Motion

If you are carrying a ball and running at
constant speed and wish to throw the ball so
that you can catch it as it comes back down,
should you (a) throw the ball at a 45o angle
above the horizontal and maintain the same
speed, (b) throw the ball straight up in the air
and slow down to catch it, or (c) throw the
ball straight up in the air and maintain the
same speed?
Chapter 3 Vectors & 2-Dimensional Motion
37
3.5 Projectile Motion

As a projectile moves in its parabolic path,
the velocity and acceleration vectors are
perpendicular to each other (a) everywhere
along its path, (b) at the peak of its path, (c)
nowhere along its path, or (d) not enough
information is given.
Chapter 3 Vectors & 2-Dimensional Motion
38
3.5 Projectile Motion

As a projectile moves in its parabolic path,
the velocity and acceleration vectors are
perpendicular to each other (a) everywhere
along its path, (b) at the peak of its path, (c)
nowhere along its path, or (d) not enough
information is given.
Chapter 3 Vectors & 2-Dimensional Motion
39
3.5 Projectile Motion

A home run is hit into the stands. The ball is
hit from home plate into the center field
stands along a parabolic path. What is the
acceleration of the ball (a) while it is rising,
(b) at the highest point of the trajectory, and
(c) while it is descending after reaching the
highest point? Ignore air resistance.
Chapter 3 Vectors & 2-Dimensional Motion
40
3.5 Projectile Motion

A home run is hit into the stands. The ball is
hit from home plate into the center field
stands along a parabolic path. What is the
acceleration of the ball (a) while it is rising,
(b) at the highest point of the trajectory, and
(c) while it is descending after reaching the
highest point? Ignore air resistance.
Chapter 3 Vectors & 2-Dimensional Motion
41
3.5 Projectile Motion

A home run is hit into the stands. The ball is
hit from home plate into the center field
stands along a parabolic path. What is the
acceleration of the ball (a) while it is rising,
(b) at the highest point of the trajectory, and
(c) while it is descending after reaching the
highest point? Ignore air resistance.
Chapter 3 Vectors & 2-Dimensional Motion
42
3.5 Projectile Motion

A home run is hit into the stands. The ball is
hit from home plate into the center field
stands along a parabolic path. What is the
acceleration of the ball (a) while it is rising,
(b) at the highest point of the trajectory, and
(c) while it is descending after reaching the
highest point? Ignore air resistance.
Chapter 3 Vectors & 2-Dimensional Motion
43
Could It Happen?
In the movie Speed a bus traveling at nearly 68
mph is rigged with a bomb that will go off if
the bus goes below 50 mph. It has to jump a
50’ gap in a bridge – could it be done?
Simple explanation
More involved Physics explanation
Vector & Projectile Motion
Practice Problems

A baseball thrown from the outfield is thrown
from shoulder height at an initial velocity of
29.4 m/s at an initial angle of 30o with respect
to the horizontal. It is in the air for a total time
interval of 3 s before it is caught by the 3rd
baseman at shoulder height level. What is the
ball’s horizontal displacement?
45
Vector & Projectile Motion
Practice Problems


A baseball thrown from the outfield is thrown
from shoulder height at an initial velocity of
29.4 m/s at an initial angle of 30o with respect
to the horizontal. It is in the air for a total time
interval of 3 s before it is caught by the 3rd
baseman at shoulder height level. What is the
ball’s horizontal displacement?
76.4 m
46
Vector & Projectile Motion
Practice Problems

A baseball thrown from the outfield is
released from shoulder height at an initial
velocity of 29.4 m/s at initial angle of 30o with
respect to the horizontal. What is the
maximum vertical displacement that the ball
reaches during its trajectory?
47
Vector & Projectile Motion
Practice Problems

A baseball thrown from the outfield is
released from shoulder height at an initial
velocity of 29.4 m/s at initial angle of 30o with
respect to the horizontal. What is the
maximum vertical displacement that the ball
reaches during its trajectory?

11.0 m
48
Vector & Projectile Motion
Practice Problems

A stone is thrown at an angle of 30o above
the horizontal from the top edge of a cliff with
an initial speed of 12 m/s. A stop watch
measures the stone’s trajectory time from the
top of the cliff to the bottom to be 5.6 s. What
is the height of the cliff?
49
Vector & Projectile Motion
Practice Problems

A stone is thrown at an angle of 30o above
the horizontal from the top edge of a cliff with
an initial speed of 12 m/s. A stop watch
measures the stone’s trajectory time from the
top of the cliff to the bottom to be 5.6 s. What
is the height of the cliff?

120 m
50
Vector & Projectile Motion
Practice Problems

A bridge that was 5 m long has been washed
out by the rain several days ago. How fast
must a car be going to successfully jump the
stream? Although the road is level on both
sides of the bridge, the road on the far side is
2 m lower than the road on this side.
51
Vector & Projectile Motion
Practice Problems

A bridge that was 5 m long has been washed
out by the rain several days ago. How fast
must a car be going to successfully jump the
stream? Although the road is level on both
sides of the bridge, the road on the far side is
2 m lower than the road on this side.

8 m/s
52
Vector & Projectile Motion
Practice Problems

A track star in the broad jump goes into the
jump at 12 m/s and launches herself at 20o
above the horizontal. How long is she in the
air before returning to the ground?
53
Vector & Projectile Motion
Practice Problems

A track star in the broad jump goes into the
jump at 12 m/s and launches herself at 20o
above the horizontal. How long is she in the
air before returning to the ground?

0.83 s
54
Vector & Projectile Motion
Practice Problems

A fireman, 50 m away from a burning
building, directs a stream of water from a
hose at an angle of 30o above the horizontal.
If the velocity of the stream of water is 40
m/s, at what height will the stream of water
strike the building?
55
Vector & Projectile Motion
Practice Problems

A fireman, 50 m away from a burning
building, directs a stream of water from a
hose at an angle of 30o above the horizontal.
If the velocity of the stream of water is 40
m/s, at what height will the stream of water
strike the building?

18.7 m
56