CHAPTER 3
TwoDimensional Motion and
Vectors
Representations:
x
y
(x, y)
(x, y)
(r,
!
)
VECTOR quantities:
Vectors have magnitude
and direction.
Other vectors: velocity,
acceleration, momentum,
force …
Vector Addition/Subtraction
•
2nd vector begins at
end of first vector
•
Order doesn’t matter
Vector addition
Vector subtraction
A – B can be interpreted
as
A+(B)
Vector Components
Cartesian components are
projections along the x
and yaxes
A
x
=
A
cos
!
A
y
=
A
sin
!
Going backwards,
A
=
A
x
2
+
A
y
2
and
!
=
tan
"
1
A
y
A
x
Example 3.1a
The magnitude of (AB) is :
a) <0
b) =0
c) >0
Example 3.1b
The xcomponent of (AB) is:
a) <0
b) =0
c) >0
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Example 3.1c
The ycomponent of (AB) > 0
a) <0
b) =0
c) >0
Example 3.2
Alice and Bob carry a bottle of wine to a
picnic site. Alice carries the bottle 5 miles due
east, and Bob carries the bottle another 10
miles traveling 30 degrees north of east.
Carol, who is bringing the glasses, takes a
short cut and goes directly to the picnic site.
How far did Carol walk?
What was Carol’s direction?
14.55 miles, at 20.10 degrees
Alice
Bob
Carol
Arcsin, Arccos and Arctan: Watch out!
same sine
same
cosine
same
tangent
Arcsin, Arccos and Arctan functions can yield
wrong angles if x or y are negative.
2dim Motion: Velocity
Graphically,
v =
"
r /
"
t
It is a vector
(rate of change of position)
Trajectory
Multiplying/Dividing Vectors by
Scalars, e.g.
"
r
/
"
t
•
Vector multiplied/divided by scalar is a vector
•
Magnitude of new vector is magnitude of
orginal vector multiplied/divided by scalar
•
Direction of new vector same as original vector
Principles of 2d Motion
•
X and Ymotion are independent
•
Two separate 1d problems
•
To get trajectory (y vs. x)
1.Solve for x(t) and y(t)
2.Invert one Eq. to get t(x)
3.Insert t(x) into y(t) to get y(x)
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 Spring '06
 Pratt
 Physics, Acceleration, Force, Momentum, Velocity, Tallahassee

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