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Ch.3 Topics. x and y parts of motion adding vectors properties of vectors projectile and circular motion relative motion. Motion in Two Dimensions. displacements: x and y parts thus: x and y velocities Ex: 30m/s North + 40m/s East = 50m/s v x + v y = v

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## 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**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**0**Two Dimensional Motion (constant acceleration)**Vector Math**• Two Methods: • geometrical (graphical) method • algebraic (analytical) method**0**Order Independent (Commutative)**0**Subtraction, head-to-head**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**0**Quadrants of x,y-Plane**0**Azimuth: Angle measured counter-clockwise from +x direction. Examples: East 0°, North 90°, West 180°, South 270°. Northeast = NE = 45°**0**Check your understanding: A: 180° B: 60° C: > 90° Note: All angles measured from east.**Point-Style Vector Notation**Example:**0**Components Example:Given A = 2.0m @ 25°, its x, y components are: Check using Pythagorean Theorem:**0**R = (2.0m, 25°) + (3.0m, 50°):**0**(cont) Magnitude, Angle:**0**General Properties of Vectors • size and direction define a vector • location independent • change size and/or direction when multiplied by a constant • written: Bold or Arrow**0**these vectors are all the same**A**0.5A -A -1.2A Multiplication by Constants 0**0**Projectile Motion • begins when projecting force ends • ends when object hits something • gravity alone acts on object**vo**Dy Dx = “Range” 0 Projectile Motion ax = 0 and ay = -9.8 m/s/s**0**Horizontal V Constant**0**Range vs. Angle**Circular Motion**• centripetal, tangential components • general acceleration vector • case of uniform circular motion**Relative Motion**• Examples: • people-mover at airport • airplane flying in wind • passing velocity (difference in velocities) • notation used:velocity “BA” = velocity of B – velocity of A**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 @ 330°. The plane’s 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)**Summary**• Vector Components & Addition using trig • Graphical Vector Addition & Azimuths • Example planar motions: Projectile Motion, Circular Motion • Relative Motion**0**Example 1: Calculate Range (R) vo = 6.00m/s qo = 30° xo = 0, yo = 1.6m; x = R, y = 0**0**Example 1 (cont.) Step 1**0**Quadratic Equation**0**Example 1 (cont.) End of Step 1**0**Example 1 (cont.) Step 2 (ax = 0) “Range” = 4.96m End of Example**0**PM Example 2: vo = 6.00m/s qo = 0° xo = 0, yo = 1.6m; x = R, y = 0**0**PM Example 2 (cont.) Step 1**0**PM Example 2 (cont.) Step 2 (ax = 0) “Range” = 3.43m End of Step 2**v1**0 1. v1 and v2 are located on trajectory. a**Q1. Given**locate these on the trajectory and form Dv. 0**0**Kinematic Equations in Two Dimensions * many books assume that xo and yo are both zero.**0**Velocity in Two Dimensions • vavg // Dr • instantaneous “v” is limit of “vavg” as Dt 0**0**Acceleration in Two Dimensions • aavg // Dv • instantaneous “a” is limit of “aavg” as Dt 0**0**Conventions • ro = “initial” position at t = 0 • r = “final” position at time t.

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