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1. Chapter 3 • Vectors and Motion in Two Dimensions

2. Major Topics • Components of Vectors • Vector Addition and Subtraction • The Acceleration Vector • Projectile Motion • Circular Motion • Relative Motion

3. Slide 3-3

4. Slide 3-4

5. Slide 3-5

6. Slide 3-6

7. Vectors A vector has both magnitude and direction Would a vector be a good quantity to represent the temperature in a room?

8. Vectors Slide 3-13

9. Coordinate systems Component Vectors

10. Components of Vectors Slide 3-22

11. Vectors have components Projections onto an orthogonal coordinate system

12. Reading Quiz  1. Ax is the __________ of the vector A. A. magnitude B. x-component C. direction D. size E. displacement Slide 3-7

13. Answer  1. Ax is the __________ of the vector A. A. magnitude B. x-component C. direction D. size E. displacement Slide 3-8

14. Checking Understanding What are the x- and y-components of these vectors? 3, 2 2, 3 3, 2 2, 3 3, 2 Slide 3-23

15. Checking Understanding What are the x- and y-components of these vectors? 3, 2 2, 3 3, 2 2, 3 3, 2 Slide 3-23

16. Checking Understanding What are the x- and y-components of these vectors? 3, 1 3, 4 3, 3 4, 3 3, 4 Slide 3-25

17. Answer What are the x- and y-components of these vectors? 3, 1 3, 4 3, 3 4, 3 3, 4 Slide 3-26

18. What is the magnitude of a vector with components (15 m, 8 m)? These bars take the magnitude of the vector argument

19. Vectors and Trigonometry The legs of a triangle depend on which angle were talking about hypotenuse opposite adjacent

20. Vectors and Trigonometry The legs of a triangle depend on which angle were talking about hypotenuse adjacent opposite

21. Vectors and Trigonometry The legs of a triangle depend on which angle were talking about hypotenuse opposite adjacent

22. Vectors and Trigonometry The legs of a triangle depend on which angle were talking about hypotenuse adjacent opposite

23. Using trig. functions SOH CAH TOA hypotenuse adjacent opposite

24. Consider the vector b⃗  with magnitude 4.00 m at an angle 23.5∘ north of east. What is the x component bx of this vector? 4 m 23.5 Degrees

25. Consider the vector b⃗  with length 4.00 m at an angle 23.5∘ north of east. What is the y component by of this vector? 4 m 23.5 Degrees

26. Checking Understanding The following vectors have length 4.0 units. What are the x- and y-components of these vectors? 3.5, 2.0 2.0, 3.5 3.5, 2.0 2.0, 3.5 3.5, 2.0 Slide 3-27

27. Answer The following vector has a length of 4.0 units. What are the x- and y-components of this vector? 3.5, 2.0 2.0, 3.5 3.5, 2.0 2.0, 3.5 3.5, 2.0 Slide 3-28

28. What is the length of the shadow cast on the vertical screen by your 10.0 cm hand if it is held at an angle of θ=30.0∘ above horizontal? light 10.0 cm hand 30 Degrees

29. What is the angle above the x axis (i.e., "north of east") for a vector with components (15 m, 8 m)?

30. Checking Understanding The following vectors have length 4.0 units. What are the x- and y-components of these vectors? 3.5, 2.0 2.0, 3.5 3.5, 2.0 2.0, 3.5 3.5, 2.0 Slide 3-29

31. Answer The following vectors have length 4.0 units. What are the x- and y-components of these vectors? 3.5, 2.0 2.0, 3.5 3.5, 2.0 2.0, 3.5 3.5, 2.0 Slide 3-30

32. Consider the two vectors C⃗  and D⃗ , defined as follows: • C⃗ =(2.35,−4.27) and D⃗ =(−1.30,−2.21). • What is the resultant vector R⃗ =C⃗ +D⃗ ? + +

33. Example Problem The labeled vectors each have length 4 units. For each vector, what is the component parallel to the ramp? The labeled vectors each have length 4 units. For each vector, what is the component perpendicular to the ramp? RAMP Slide 3-31

34. Example Problem The labeled vectors each have length 4 units. For each vector, what is the component parallel to the ramp? The labeled vectors each have length 4 units. For each vector, what is the component perpendicular to the ramp? Slide 3-31

35. Example Problem The labeled vectors each have length 4 units. For each vector, what is the component parallel to the ramp? The labeled vectors each have length 4 units. For each vector, what is the component perpendicular to the ramp? Slide 3-31

36. Example Problem The labeled vectors each have length 4 units. For each vector, what is the component parallel to the ramp? The labeled vectors each have length 4 units. For each vector, what is the component perpendicular to the ramp? Slide 3-31

37. Example Problem The labeled vectors each have length 4 units. For each vector, what is the component parallel to the ramp? The labeled vectors each have length 4 units. For each vector, what is the component perpendicular to the ramp? Slide 3-31

38. Example Problems The Manitou Incline was an extremely steep cog railway in the Colorado mountains; cars climbed at a typical angle of 22 with respect to the horizontal. What was the vertical elevation change for the one-mile run along the track? 22 Slide 3-32

39. Example Problems • The maximum grade of interstate highways in the United States is 6.0%, meaning a 6.0 meter rise for 100 m of horizontal travel. • What is the angle with respect to the horizontal of the maximum grade? Slide 3-32

40. Example Problems • The maximum grade of interstate highways in the United States is 6.0%, meaning a 6.0 meter rise for 100 m of horizontal travel. • What is the angle with respect to the horizontal of the maximum grade? • Suppose a car is driving up a 6.0% grade on a mountain road at 67 mph (30m/s). How many seconds does it take the car to increase its height by 100 m? Find displacement . Slide 3-32

41. Example Problems • The maximum grade of interstate highways in the United States is 6.0%, meaning a 6.0 meter rise for 100 m of horizontal travel. • What is the angle with respect to the horizontal of the maximum grade? • Suppose a car is driving up a 6.0% grade on a mountain road at 67 mph (30m/s). How many seconds does it take the car to increase its height by 100 m? OR Find displacement . Slide 3-32

42. Vector Addition When adding vectors, bring the tip of one to the tail of the other

43. Application of vector addition 2D Throw a ball up while moving on the motorcycle Speed of ball relative to ground y(meters) ? 10 m/s 2 m/s x(meters) 5 10 Use the Pythagorean Theorem =

44. What is the ball’s speed? Solve for c = 2 m/s 10 m/s ~ 10.2

45. Checking Understanding Which of the vectors below best represents the vector sum P + Q?   Slide 3-16

46. Answer Which of the vectors below best represents the vector sum P + Q?   A. Slide 3-17

47. Answer Which of the vectors below best represents the vector sum P + Q?   Slide 3-17

48. Slide 3-14

49. Vector Subtraction Flip this vector