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Chapter 3: Motion in a Plane

Chapter 3: Motion in a Plane. Vector Addition Velocity Acceleration Projectile motion Relative Velocity CQ: 1, 2. P: 3, 5, 7, 13, 21, 31, 39, 49, 51. Two Dimensional Vectors. Displacement, velocity, and acceleration each have (x, y) components Two methods used:

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Chapter 3: Motion in a Plane

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  1. Chapter 3: Motion in a Plane • Vector Addition • Velocity • Acceleration • Projectile motion • Relative Velocity • CQ: 1, 2. • P: 3, 5, 7, 13, 21, 31, 39, 49, 51.

  2. Two Dimensional Vectors Displacement, velocity, and acceleration each have (x, y) components Two methods used: geometrical (graphical) method algebraic (analytical) method / 2

  3. Graphical, Tail-to-Head

  4. Addition Example • Giam (11)

  5. 0 Order Independent (Commutative)

  6. 0 Subtraction, tail-to-tail

  7. Subtraction Example • Giam (19)

  8. 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

  9. Vector Components

  10. Examples • Magnitude || (g4-5) Notation, Example • Component Example Animated • Phet Vectors

  11. h o q a 0 Trigonometry

  12. 0 Using your Calculator: Degrees and Radians Check this to verify your calculator is working with degrees

  13. h o q a Example: 0 • Given: • = 10°, h = 3 Find o and a.

  14. Determine angle from length ratios. Ex. o/h = 0.5: Ex. o/a = 1.0: 0 Inverse Trig

  15. h o q a 0 Pythagorean Theorem Example: Given, o = 2 and a = 3 Find h

  16. 0 Azimuth: Angle measured counter-clockwise from +x direction. Examples: East 0°, North 90°, West 180°, South 270°. Northeast = NE = 45°

  17. 0 Check your understanding: What are the Azimuth angles? A: 180° 60° B: 70° C: 110° Note: All angles measured from east.

  18. 0 Components: Given A = 2.0m @ 25°, its x, y components are: Check using Pythagorean Theorem:

  19. Vector Addition by Components: 0

  20. Example Vector Addition 0 R = (10cm, 0°) + (10cm, 45°):

  21. 0 (cont) Magnitude, Angle:

  22. 0 General Properties of Vectors • size and direction define a vector • location independent • change size and/or direction when multiplied by a constant • Vector multiplied by a negative number changes to a direction opposite of its original direction. • written: Bold or Arrow

  23. 0 these vectors are all the same

  24. A 0.5A -A -1.2A Multiplication by Constants 0

  25. Projectile Motion • time = 0: e.g. baseball leaves fingertips • time = t: e.g. baseball hits glove • Horizontal acceleration = 0 • Vertical acceleration = -9.8m/s/s • Horizontal Displacement (Range) = Dx • Vertical Displacement = Dy • Vo = launch speed • qo = launch angle

  26. 0 Range vs. Angle

  27. Example 1: 6m/s at 30 0 vo = 6.00m/s qo = 30° xo = 0, yo = 1.6m; x = R, y = 0 27

  28. Example 1 (cont.) 0 Step 1 28

  29. Quadratic Equation 0 29

  30. Example 1 (cont.) 0 End of Step 1 30

  31. Example 1 (cont.) 0 Step 2 (ax = 0) “Range” = 4.96m End of Example 31

  32. 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

  33. Summary • Vector Components & Addition using trig • Graphical Vector Addition & Azimuths • Projectile Motion • Relative Motion

  34. 0 R = (2.0m, 25°) + (3.0m, 50°): 34

  35. 0 (cont) Magnitude, Angle: 35

  36. 0 PM Example 2: vo = 6.00m/s qo = 0° xo = 0, yo = 1.6m; x = R, y = 0

  37. 0 PM Example 2 (cont.) Step 1

  38. 0 PM Example 2 (cont.) Step 2 (ax = 0) “Range” = 3.43m End of Step 2

  39. PM Example 2: Speed at Impact

  40. v1 0 1. v1 and v2 are located on trajectory. a

  41. Q1. Given locate these on the trajectory and form Dv. 0

  42. 0 Kinematic Equations in Two Dimensions * many books assume that xo and yo are both zero.

  43. 0 Velocity in Two Dimensions • vavg // Dr • instantaneous “v” is limit of “vavg” as Dt  0

  44. 0 Acceleration in Two Dimensions • aavg // Dv • instantaneous “a” is limit of “aavg” as Dt  0

  45. 0 Conventions • ro = “initial” position at t = 0 • r = “final” position at time t.

  46. Dr ro r 0 Displacement in Two Dimensions

  47. Acceleration ~ v change • 1 dim. example: car starting, stopping

  48. Acceleration, Dv, in Two Dimensions 0

  49. Ex. Vector Addition • Add A = 3@60degrees azimuth, plus B = 3@300degrees azimuth. • Find length of A+B, and its azimuth. Sketch the situation.

  50. Ex.2: • 10cm@10degrees + 10cm@30degrees • Length and azimuth?

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