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Chapter 3

Chapter 3. Vectors and Motion in Two Dimensions. Major Topics. Components of Vectors Vector Addition and Subtraction The Acceleration Vector Projectile Motion Circular Motion Relative Motion. 3 Vectors and Motion in Two Dimensions. Slide 3-2. Slide 3-3. Slide 3-4. Slide 3-5.

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Chapter 3

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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. 3Vectors and Motion in Two Dimensions Slide 3-2

  4. Slide 3-3

  5. Slide 3-4

  6. Slide 3-5

  7. Slide 3-6

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

  9. Vectors Slide 3-13

  10. Coordinate systems Component Vectors

  11. Components of Vectors Slide 3-22

  12. Vectors have components Projections onto an orthogonal coordinate system

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

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

  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, 2 2, 3 3, 2 2, 3 3, 2 Slide 3-23

  17. 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

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

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

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

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

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

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

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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

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

  31. 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

  32. 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

  33. 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 ? + +

  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? 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 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

  39. 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

  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? 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? Find displacement . Slide 3-32

  42. 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

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

  44. 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 =

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

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

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

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

  49. Slide 3-14

  50. Vector Subtraction Flip this vector

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