Vectors in 2D and Projectile Motion

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Vectors in 2D and Projectile Motion
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Vectors in 2D and Projectile Motion
Recall our definition of a vectorsomething that
had both a magnitude (of some sort) and a
direction. Examples displacement, velocity,
acceleration.
N
35 north of East
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Vectors in 2D and Projectile Motion
Recall our definition of a vectorsomething that
had both a magnitude (of some sort) and a
direction. Examples displacement, velocity,
acceleration. Vectors add tip to tail
A B
B
A
N
35 north of East
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Vectors in 2D and Projectile Motion
Recall our definition of a vectorsomething that
had both a magnitude (of some sort) and a
direction. Examples displacement, velocity,
acceleration. Vectors add tip to tail and -B
is opposite in direction to B (but with the same
magnitude).
A B
B
A
-B
N
35 north of East
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We often are interested in the components (or
component vectors) of a vector. These are two
perpendicular vectors that add up to the original
vector
A
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We often are interested in the components (or
component vectors) of a vector. These are two
perpendicular vectors that add up to the original
vector
A
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We often are interested in the components (or
component vectors) of a vector. These are two
perpendicular vectors that add up to the original
vector
y
A
Ay
Ax
x
Coordinate axes (x, y) are usually chosen as
directions for the component vectors, written Ax
and Ay.
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We often are interested in the components (or
component vectors) of a vector. These are two
perpendicular vectors that add up to the original
vector
y
A
Ay
Ax Ay A
Ax
x
Coordinate axes (x, y) are usually chosen as
directions for the component vectors, written Ax
and Ay.
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Elementary trigonometry allows us to express the
component vectors in terms of vector A (its
magnitude and direction).
y
A
Ay
Ax Ay A
?
Ax
x
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Note Ax is positive if the vector points in the
x direction Ay is positive if it points in the
y direction
y
A
Ay
?
Ax
x
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How we write the component vectors depends on
what angle we are using to indicate direction
y
Ax
x
f
Ay
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y
Ax
x
?
Ay
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y
Ax
x
f
Ay
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y
Ax
x
?
Ay
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Putting this vector information to use
projectile motion in 2Dwhat happens when you
throw or kick something into the air?
vi
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Putting this vector information to use
projectile motion in 2Dwhat happens when you
throw or kick something into the air?
up down
vi
sideways


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Putting this vector information to use
projectile motion in 2Dwhat happens when you
throw or kick something into the air?
up down
vi
sideways


Lets consider a simpler case to start off
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
vi
1 m
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
vi
1 m
Motion occurs along two independent,
perpendicular directions (the vertical and the
horizontal directionslets call them y and x).
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
vi
1 m
Motion occurs along two independent,
perpendicular directions (the vertical and the
horizontal directionslets call them y and
x). Where should we place the coordinate origin?
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
vi
1 m
Motion occurs along two independent,
perpendicular directions (the vertical and the
horizontal directionslets call them y and
x). Where should we place the coordinate origin?
(It doesnt change the motion but some locations
are more convenient than others for analyzing the
motion.)
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
vi
x
1 m
y
One reasonable choice
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
y
vi
1 m
x
Another reasonable choice
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
y
Lets choose this one (we have to start with one
or the other)
vi
1 m
x
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
y
Lets choose this one (we have to start with one
or the other)
vi
1 m
x
Starting information xi 0 m and yi 1m
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
y
Lets choose this one (we have to start with one
or the other)
vi
1 m
x
Starting information xi 0 m and yi
1m (vx)i 1 m/s and (vy)i 0 m/s
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
y
Lets choose this one (we have to start with one
or the other)
vi
1 m
x
Starting information xi 0 m and yi
1m (vx)i 1 m/s and (vy)i 0 m/s ax 0 m/s2
and ay -9.8 m/s2
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A cup gets knocked off the edge of a 1 m high
table. Lets assume it has initial velocity vi
of 1 m/s at the beginning of its fall
y
Lets choose this one (we have to start with one
or the other)
t 0
vi
1 m
x
Starting information xi 0 m and yi
1m (vx)i 1 m/s and (vy)i 0 m/s ax 0 m/s2
and ay -9.8 m/s2
Lets start the stopwatch that keeps track of
time when the cups first gets knocked.
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What we would like to describe is where the cup
is during its fall, at what time.
y
t 0
vi 1 m/s in the x direction
1 m
x
In answering this we are helped by the fact that
the horizontal motion (x axis) is independent of
the vertical motion (y axis). They are
connected in a way, but much of the analysis for
one may be done without thinking about the other!!
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The vertical motion controls how long the cup is
in the air. The horizontal motion controls how
far the cup moves sideways during its fall. Lets
do the problem on the board