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 3

4. Addition Example Giam (11)

5. Order Independent (Commutative) 5

6. Subtraction, tail-to-tail 6

7. Subtraction Example Giam (19)

8. Algebraic Component Addition trigonometry & geometry “R” denotes “resultant” sum R x = sum of x-parts of each vector R y = sum of y-parts of each vector 8

9. Vector Components 9

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

11. Trigonometry  o a h 11

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

13. Example:  o a h Given:  = 10°, h = 3 Find o and a. 13

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

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

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

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

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

19. Vector Addition by Components: 19

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

21. (cont) Magnitude, Angle: 21

22. 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 22

23. these vectors are all the same 23

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

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) =  x Vertical Displacement =  y V o = launch speed  o = launch angle

26. Range vs. Angle 26

27. Example 1: 6m/s at 30 v o = 6.00m/s  o = 30° x o = 0, y o = 1.6m; x = R, y = 0 27

28. Example 1 (cont.) Step 1 28

29. Quadratic Equation 29

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

31. Example 1 (cont.) Step 2 (a x = 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 32

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

34. R = (2.0m, 25°) + (3.0m, 50°): 34

35. (cont) Magnitude, Angle: 35

36. PM Example 2: v o = 6.00m/s  o = 0° x o = 0, y o = 1.6m; x = R, y = 0 36

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

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

39. PM Example 2: Speed at Impact 39

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

41. Q1. Givenlocate these on the trajectory and form  v. 41

42. Kinematic Equations in Two Dimensions * many books assume that x o and y o are both zero. 42

43. Velocity in Two Dimensions v avg //  r instantaneous “v” is limit of “v avg ” as  t  0 43

44. Acceleration in Two Dimensions a avg //  v instantaneous “a” is limit of “a avg ” as  t  0 44

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

46. Displacement in Two Dimensions roro r rr 46

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

48. Acceleration, Dv, in Two Dimensions 48

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?

51. Calculate F3 F1 = 8@60, F2 = 5.5@-45 F1 + F2 + F3 = 0 F3 = -(F1 + F2) Rx = 8cos60+5.5cos(-45)=7.89 Ry = 8sin60+5.5sin(-45)=3.04 R = 8.46 Angle = tan-1(3.04/7.89) = 21 deg above +x axis Answer book wants is 180 off this angle!

52. Addition by Parts (Components) 52