Ch.3 Topics

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Ch.3 Topics

• x and y parts of motion
• adding vectors
• properties of vectors
• projectile and circular motion
• relative motion

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Motion in Two Dimensions

• displacements x and y parts
• thus x and y velocities
• Ex
• 30m/s North 40m/s East 50m/s
• vx vy v
• component set vector

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Two Dimensional Motion (constant acceleration)
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Vector Math

• Two Methods
• geometrical (graphical) method
• algebraic (analytical) method

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Graphical, Tail-to-Head
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Order Independent (Commutative)
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Subtraction, head-to-head
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Example Subtraction Dv.
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Algebraic Component Addition

• trigonometry geometry
• R denotes resultant sum
• Rx sum of x-parts of each vector
• Ry sum of y-parts of each vector

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Addition by Parts (Components)
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Vector Components
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Quadrants of x,y-Plane
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Azimuth
Angle measured counter-clockwise from x
direction.
Examples East 0, North 90, West 180, South
270. Northeast NE 45
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Check your understanding
A 180 B 60 C gt 90
Note All angles measured from east.
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Unit Vectors, i, j, k
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Point-Style Vector Notation
Example
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Components ExampleGiven A 2.0m _at_ 25, its x,
y components are
Check using Pythagorean Theorem
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Vector Addition by Components
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R (2.0m, 25) (3.0m, 50)
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(cont) Magnitude, Angle
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General Properties of Vectors
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• size and direction define a vector
• location independent
• change size and/or direction when multiplied by a
  constant
• written Bold or Arrow
  

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these vectors are all the same
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Multiplication by Constants
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Projectile Motion
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• begins when projecting force ends
• ends when object hits something
• gravity alone acts on object

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Projectile Motion
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ax 0 and ay -9.8 m/s/s
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Horizontal V Constant
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Range vs. Angle
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Circular Motion

• centripetal, tangential components
• general acceleration vector
• case of uniform circular motion

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Relative Motion

• Examples
• people-mover at airport
• airplane flying in wind
• passing velocity (difference in velocities)
• notation usedvelocity BA velocity of B
  velocity of A

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Example
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Ex. A Plane has an air speed vpa 75m/s. The
wind has a velocity with respect to the ground of
vag 8 m/s _at_ 330. The planes path is due North
relative to ground. a) Draw a vector diagram
showing the relationship between the air speed
and the ground speed. b) Find the ground speed
and the compass heading of the plane.
(similar situation)
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Summary

• Vector Components Addition using trig
• Graphical Vector Addition Azimuths
• Example planar motions Projectile Motion,
  Circular Motion
• Relative Motion

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Example 1 Calculate Range (R)
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vo 6.00m/s qo 30 xo 0, yo 1.6m x
R, y 0
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Example 1 (cont.)
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Step 1
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Quadratic Equation
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Example 1 (cont.)
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End of Step 1
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Example 1 (cont.)
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Step 2 (ax 0)
Range 4.96m
End of Example
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PM Example 2
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vo 6.00m/s qo 0 xo 0, yo 1.6m x
R, y 0
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PM Example 2 (cont.)
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Step 1
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PM Example 2 (cont.)
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Step 2 (ax 0)
Range 3.43m
End of Step 2
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PM Example 2 Speed at Impact
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1. v1 and v2 are located on trajectory.
a
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Q1. Given
locate these on the trajectory and form Dv.
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Kinematic Equations in Two Dimensions
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many books assume that xo and yo are both zero.
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Velocity in Two Dimensions
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• vavg // Dr
• instantaneous v is limit of vavg as Dt ? 0

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Acceleration in Two Dimensions

• aavg // Dv
• instantaneous a is limit of aavg as Dt ? 0

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Conventions
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• ro initial position at t 0
• r final position at time t.

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Displacement in Two Dimensions
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Acceleration v change

• 1 dim. example car starting, stopping

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Acceleration, Dv, in Two Dimensions
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1. v1 and v2 are located on trajectory.
a
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Ex. If v1(0.00s) 12m/s, 60 and v2(0.65s)
7.223 _at_ 33.83, find aave.
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Q1. Given
locate these on the trajectory and form Dv.
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Q2. If v3(1.15s) 6.06m/s, -8.32 and v4(1.60s)
7.997, -41.389, write the coordinate-forms of
these vectors and calculate the average
acceleration.