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Chapter 3: Motion in a Plane

This chapter explores vector addition, velocity, acceleration, projectile motion, and relative velocity in two dimensions. Learn geometric and algebraic methods, trigonometry, and vector properties. Understand projectile motion, relative motion, and kinematic equations in two dimensions. Develop knowledge on displacement, velocity, and acceleration components, and their applications in various scenarios.

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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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