Kinematics in Two Dimensions

1. Kinematics in Two Dimensions

2. Section 1: Adding Vectors Graphically

3. Adding Vectors Graphically • Remember vectors have magnitude (length) and direction. • When you add vectors you must maintain both magnitude and direction • This information is represented by an arrow (vector)

4. A vector has a magnitude and a direction • The length of a drawn vector represents magnitude. • The arrow represents the direction Larger Vector Smaller Vector

5. Graphical Representation of Vectors • Given Vector a: Draw 2a Draw -a

6. Problem set 1: • Which vector has the largest magnitude? • What would -b look like? • What would 2 c look like? a c b

7. Vectors • Three vectors a c b

8. a c b • When adding vectors graphically, align the vectors head-to-tail. • This means draw the vectors in order, matching up the point of one arrow with the end of the next, indicating the overall direction heading. • Ex. a + c • The starting point is called the origin c a origin

9. a c b • When all of the vectors have been connected, draw one straight arrow from origin to finish. This arrow is called the resultant vector. c a origin

10. a c b • Ex.1 Draw a + b

11. a c b • Ex.1 Draw a + b Resultant origin

12. a c b • Ex. 2 Draw a + b + c

13. a c b • Ex. 2 Draw a + b + c Resultant origin

14. a c b • Ex. 3 Draw 2a – b – 2c

15. a c b • Ex. 3 Draw 2a – b – 2c origin Resultant

16. Section 2: How do you name vector directions?

17. Vector Direction Naming • How many degrees is this? N W E S

18. Vector Direction Naming • How many degrees is this? N 90º W E S

19. Vector Direction Naming • What is the difference between 15º North of East and 15 º East of North? N W E S

20. Vector Direction Naming • What is the difference between 15º North of East and 15º East of North? (can you tell now?) N N W E W E S S 15º North of East 15º East of North

21. Vector Direction Naming N 15º W S 15º North of what?

22. Vector Direction Naming N 15º W E S 15º North of East

23. 15º W E S 15º East of What?

24. N 15º W E S 15º East of North

25. ___ of ___ N E This is the baseline. It is the direction you look at first This is the direction you go from the baseline to draw your angle

26. Describing directions • 30º North of East • East first then 30º North • 40º South of East • East first then 30º South • 25º North of West • West first then 30º North • 30º South of West • West first then 30º South

27. Problem Set #2 (Name the angles) 30º 45º 20º 30º 20º

28. Intro: Get out your notes b • Draw the resultant of a – b + c 2. What would you label following angles a. b. 3. Draw the direction 15º S of W a c 28º 18º

30. Section 3: How do you add vectors mathematically (not projectile motion)

31. The Useful Right Triangle • Sketch a right triangle and label its sides c: hypotenuse a: opposite Ө b: adjacent The angle

32. The opposite (a) and adjacent (b) change based on the location of the angle in question • The hypotenuse is always the longest side Ө c: hypotenuse b: adjacent a: opposite

33. The opposite (a) and adjacent (b) change based on the location of the angle in question • The hypotenuse is always the longest side Ө c: hypotenuse b: adjacent a: opposite

34. To figure out any side when given two other sides • Use Pythagorean Theorem a2 + b2 =c2 c: hypotenuse a: opposite Ө b: adjacent The angle

35. Sometimes you need to use trig functions c: hypotenuse a: opposite Ө a: adjacent Opp Hyp Opp Adj Sin Ө = _____ Tan Ө = _____ Adj Hyp Cos Ө = _____

36. Sometimes you need to use trig functions c: hypotenuse a: opposite Ө a: adjacent Opp Hyp Opp Adj Sin Ө = _____ Tan Ө = _____ SOH CAH TOA Adj Hyp Cos Ө = _____

37. More used versions Opp Hyp Sin Ө = _____ Opp = (Sin Ө)(Hyp) Adj Hyp Cos Ө = _____ Adj = (Cos Ө)(Hyp) Opp Adj Opp Adj Ө = Tan-1 _____ Tan Ө = _____

38. To resolve a vector means to break it down into its X and Y components. Example: 85 m 25º N of W • Start by drawing the angle 25º

39. To resolve a vector means to break it down into its X and Y components. Example: 85 m 25º N of W • Start by drawing the angle • The magnitude given is always the hypotenuse 85 m 25º

40. To resolve a vector means to break it down into its X and Y components. Example: 85 m 25º N of W • this hypotenuse is made up of a X component (West) • and a Y component (North) 85 m • North 25º West

41. In other words: I can go so far west along the X axis and so far north along the Y axis and end up in the same place finish finish 85 m • North origin origin 25º West

42. If the question asks for the West component: Solve for that side • Here the west is the adjacent side Adj = (Cos Θ)(Hyp) 85 m 25º West or Adj.

43. If the question asks for the West component: Solve for that side • Here the west is the adjacent side Adj = (Cos Θ)(Hyp) Adj = (Cos 25º)(85) = 77 m W 85 m 25º West or Adj.

44. If the question asks for the North component: Solve for that side • Here the north is the opposite side Opp = (Sin Θ)(Hyp) 85 m • North • or • Opp. 25º

45. If the question asks for the North component: Solve for that side • Here the west is the opposite side Opp = (Sin Θ)(Hyp) Opp = (Sin 25º)(85) = 36 m N 85 m • North • or • Opp 25º

46. Resolving Vectors Into Components • Ex 4a. Find the west component of 45 m 19º S of W

47. Resolving Vectors Into Components • Ex 4a. Find the west component of 45 m 19º S of W

48. Ex 4a. Find the south component of 45 m 19º S of W

49. Ex 4a. Find the south component of 45 m 19º S of W

50. Remember the wording. These vectors are at right angles to each other. 5 m/s forward 5 m/s Redraw and it becomes 30 m/s Hypotenuse = Resultant speed velocity = 30 m/s down Right angle