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Chapter 2: Vectors
Vector: a quantity which has magnitude (how big) and direction
Vectors
displacement
velocity
acceleration
force
Scalars
distance
speed
Vectors are denote by bold face or arrows
V or V
The magnitude of a vector is denoted by plain text
V
Vectors can be graphically represented by arrows
direction
magnitude
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Vector Addition: Graphical Methodof R = A + B
•Shift B parallel to itself until its tail is at the head of A,
retaining its original length and direction.
•Draw R (the resultant) from the tail of A to the head of B.
B
B
A
+
= A
=
R
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the order of addition of several vectors does not matter
B
C
D
A
C
C
D
A
B
B
D
A
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If the vectors to be added are perpendicular to each other, then
trigonometric methods can easily be applied
opposite
A
sin q 

hypotenuse C
C
A
q
B
adjacent
B
cosq 

hypotenuse C
opposite A
tan q 

adjacent B
A2  B2  C 2
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Vector Subtraction: the negative of a vector points in the opposite
direction, but retains its size (magnitude)
• A- B = A +( -B)
A
B
-
= A
-B
R
=

-B
A
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Resolving a Vector
replacing a vector with two or more (mutually perpendicular)
vectors => components
directions of components determined by coordinates or
geometry.
B
C
=
A
+
Examples:
horizontal and vertical
North-South and East-West
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Example: As a youth, Dr. Gallis walked 6 km north north east (60º
north of east) to reach school. What are the East and North
“components” of his displacement?
N
y
D
D
y
q = 60º
x
Dx
E
Example 2.3: A woman on the ground sees an airplane climbing at
an angle of 35º above the horizontal. By driving at 70 mph, she is
able to stay directly below the airplane. What is the speed of the
airplane?
y
v
q = 35º
x
vx =70 mph, horizontal
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Vector Addition by components
R=A+B+C
Resolve vectors into components(Ax, Ay etc. )
Add like components
Ax + Bx + Cx = Rx
Ay + By + Cy = Ry
The magnitude and direction of the resultant R can be
determined from its components.
Example 2.5: The sailboat Ardent Spirit is headed due north at a
forward speed of 6.0 knots (kn). The pressure of the wind on its sails
causes the boat to move sideways to the east at 0.5 kn. A tidal current
is flowing to the southwest at 3.0 kn. What is the velocity of Ardent
Spirit relative to the earth’s surface?
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Motion in the vertical plane
idealized horizontal motion: no acceleration
idealized vertical motion: acceleration of gravity (downwards)
Motion in the vertical plane is a combination of these two motions
motion is resolved into horizontal and vertical components
Compare motion of a dropped ball and a ball rolled off of a
horizontal table
v v v 0 v 0 v 0
Ax
0
Ay
Bx
By
xB  0
x A  v0t
1 2
y  gt
2
v y  gt
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Example: A ball is rolled off of a 1m high horizontal table with an
initial speed of 5 m/s.
How long is the ball in the air?
How far from the table does the ball land?
What is the final velocity of the ball?
What is the final speed of the ball?
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Projectile Flight
object with an initial velocity v0 at an angle of q with respect to
the horizon
v0
v0 x  v 0 cosq
v0 y  v0 sin q
v x  v0 x
v y  v 0 y - gt
x  x 0  v0 x t
q
Initial velocity components
Velocity components at later times
1 2
y  y0  v 0 y t - gt
2
Position of projectile
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Example 2.7: A ball is thrown at 20 m/s at an angle of 65º above
the horizontal. The ball leaves the thrower’s hand at a height of
1.8 m. At what height will it strike a wall 10 m away?
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v0
Projectile Range
q
R
2v0 y
1 2
y  y0  v 0 y t - gt  y 0 at impact  t 
2
g
2v0 y
R  x - x 0  v0 x t  v 0 x
g
2v 0 sin q
v0 2
 v0 cosq
2
sin q cosq
g
g
v0 2
R
sin 2q
g
R max
v0 2

g
q 2  90 - q 1 (angles with the same range)
(at 45º )
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Example 2.8: An arrow leaves a bow at 30 m/s.
What is its maximum range?
At what two angles could the archer point the arrow for a target
70 m away?
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