1. Ch.3 Topics • x and y parts of motion • adding vectors • properties of vectors • projectile and circular motion • relative motion

2. 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

3. 0 Two Dimensional Motion (constant acceleration)

4. Vector Math • Two Methods: • geometrical (graphical) method • algebraic (analytical) method

5. Graphical, Tail-to-Head

6. 0 Order Independent (Commutative)

7. 0 Subtraction, head-to-head

8. Example Subtraction: Dv.

9. 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

10. Addition by Parts (Components) 0

11. Vector Components

12. 0 Quadrants of x,y-Plane

13. 0 Azimuth: Angle measured counter-clockwise from +x direction. Examples: East 0°, North 90°, West 180°, South 270°. Northeast = NE = 45°

14. 0 Check your understanding: A: 180° B: 60° C: > 90° Note: All angles measured from east.

15. Unit Vectors, i, j, k

17. Point-Style Vector Notation Example:

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

19. Vector Addition by Components: 0

20. 0 R = (2.0m, 25°) + (3.0m, 50°):

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 • written: Bold or Arrow

23. 0 these vectors are all the same

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

25. 0 Projectile Motion • begins when projecting force ends • ends when object hits something • gravity alone acts on object

26. vo Dy Dx = “Range” 0 Projectile Motion ax = 0 and ay = -9.8 m/s/s

27. 0 Horizontal V Constant

28. 0 Range vs. Angle

29. Circular Motion • centripetal, tangential components • general acceleration vector • case of uniform circular motion

30. 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

31. Example:

32. 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)

34. Summary • Vector Components & Addition using trig • Graphical Vector Addition & Azimuths • Example planar motions: Projectile Motion, Circular Motion • Relative Motion

36. 0 Example 1: Calculate Range (R) vo = 6.00m/s qo = 30° xo = 0, yo = 1.6m; x = R, y = 0

37. 0 Example 1 (cont.) Step 1

38. 0 Quadratic Equation

39. 0 Example 1 (cont.) End of Step 1

40. 0 Example 1 (cont.) Step 2 (ax = 0) “Range” = 4.96m End of Example

41. 0 PM Example 2: vo = 6.00m/s qo = 0° xo = 0, yo = 1.6m; x = R, y = 0

42. 0 PM Example 2 (cont.) Step 1

43. 0 PM Example 2 (cont.) Step 2 (ax = 0) “Range” = 3.43m End of Step 2

44. PM Example 2: Speed at Impact

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

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

47. 0 Kinematic Equations in Two Dimensions * many books assume that xo and yo are both zero.

48. 0 Velocity in Two Dimensions • vavg // Dr • instantaneous “v” is limit of “vavg” as Dt  0

49. 0 Acceleration in Two Dimensions • aavg // Dv • instantaneous “a” is limit of “aavg” as Dt  0

50. 0 Conventions • ro = “initial” position at t = 0 • r = “final” position at time t.