Two-dimensional motion

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Two-dimensional motion
Curved motion -or- straight-line motion that is
not aligned with a single axis.
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Position
Position is the (x,y) coordinates of an object on
a 2-D Cartesian coordinate system. x is the
x-position. y is the y-position.
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Distance
Distance (from the origin) is found using the
Pythagorean theorem.
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Angle
Memorize the following trig functions.
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Angle (Example)
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Converting distance and angle to position
(coordinates)
A child pulls a sled with a 2-m long rope that
makes an angle of 30 with respect to the ground.
If the origin is at the point where the rope is
tied to the sled, what is the position of the
childs hand?
-Sketch a picture. -Draw the right triangle
showing the x and y components and the given
angle. -Use trig functions to calculate the x and
y components (i.e. legs of the right triangle).
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Poll
The position of an object is at x 3 m and y
?4 m. Which arrow most closely points in the
direction of the objects position?
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Poll
Your table in the physics lab is 8.5 m from a
certain corner of the room at an angle of 210?
from East. What is the position of your table?
(Use N as the y axis and E as the x axis).

1. x ?7.36 m, y ?4.25 m
2. x 7.36 m, y 4.25 m
3. x ?4.25 m, y 7.36 m
4. x 4.25 m, y ?7.36 m
5. x ?7.36 m, y 4.25 m
6. x ?7.36 m, y ?4.25 m

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Poll
The center of the bar, relative to a pole vaulter
getting ready to run, is at the position x 15
m, y5 m. At what distance is the bar, relative
to the pole vaulter?

1. 15 m
2. 20 m
3. 15.8 m
4. 250 m
5. 10 m

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Displacement
Displacement is the vector (i.e. arrow) from an
initial position to a final position. Displacement
tells you direction and distance an object
moved. Displacement is the change in position
of an object.
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Example
What is the displacement of the toy car? Sketch
the vector and write the x-displacement and
y-displacement. What distance does it travel and
at what angle?
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Poll
A Volkswagon moves from position 1 to position 2.
Which arrow most closely points in the direction
of the displacement of the car?
t1
t2
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Poll
A soccer ball is displaced ?x ?14 m, ?y 4 m
where x is to the right and y is forward
toward the goal. The displacement vector tells
us that the soccer ball traveled

1. to the right and toward the goal
2. to the left and toward the goal
3. to the left and away from the goal
4. to the left and toward the goal

5. straight to the right 6. straight to the
left 7. straight toward the goal 8. straight away
from the goal
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Total Displacement
A soccer ball is kicked with a displacement ?x
?14 m, ?y 4 m along the ground. Its then
kicked again with the displacement ?x 4 m, ?y
8 m. What is the total displacement of the ball
from its original position?
To add vectors algebraically, add their
components separately. The resulting vector is
called the resultant. To add them pictorially,
draw them tail to head, one after the other. The
resultant is from the tail of the first vector to
the head of the last vector.
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Example
A car travels from x 4, y 4 to x ?4, y4
and then to x ?3, y ?3. How far is the car
from where it started?
Chapter 1 Vectors
Matter Interactions PHY 221 Dr. Aaron Titus
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Poll
A car travels from position 1 to position 2 to
position 3. Which is larger, the distance
traveled or the magnitude of the total
displacement?

1. distance traveled
2. magnitude of total displacement
3. neither, because they are equal

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Average Velocity
In the same direction as displacement. Time
interval may be large.
Intantaneous Velocity
In the same direction as displacement during a
small time interval.
Instantaneous velocity is ALWAYS tangent to the
path and in the direction of motion.
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Instantaneous Velocity
A pendulum swings back and forth. What is the
pendulums direction of motion at the lowest
point of its swing, as it travels from right to
left?
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Inst. velocity is tangent to the path
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Poll
A comet orbits Sun counterclockwise as shown
below. What is the direction of average velocity
of the comet between points A and B?
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Poll
A comet orbits Sun counterclockwise as shown
below. What is the direction of intantaneous
velocity of the comet between at point A?
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Poll
A baseball is thrown with an x-velocity of 30 m/s
and a y-velocity of -2 m/s. The ball is traveling

1. to the right and downward
2. to the left and downward
3. to the right and upward
4. to the left and upward
5. straight to the left
6. straight to the right
7. straight downward
8. straight upward

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Speed is magnitude of instantaneous velocity
vx
vy
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Poll
A high jumper leaves the ground with a speed of 8
m/s at an angle of 70? with respect to the
ground. What is his x-velocity and y-velocity
when he leaves the ground?

1. vx 8 m/s, vy 0 m/s
2. vx 2.74 m/s, vy 5.72 m/s
3. vx 5.72 m/s, vy 2.74 m/s
4. vx 8 m/s, vy 8 m/s
5. none of the above

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Average Acceleration
As an object travels along a curve, its velocity
vector will change in direction (and maybe in
magnitude). The change in velocity per second is
the average acceleration.
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Poll
What direction is the average acceleration during
the interval shown?

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Example
A football player is running with a velocity lt 5,
5gt m/s when he is hit by a defensive player. The
collision lasts 1.5 s. His velocity after the
collision is lt 2, 0 gt m/s. What is the players
average acceleration during the collision?