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Chapter 4 Lecture

Chapter 4 Lecture. Chapter 4 Kinematics in Two Dimensions. Chapter Goal: To learn how to solve problems about motion in a plane. Slide 4-2. Chapter 4 Preview. Slide 4-3. Chapter 4 Preview. Slide 4-4. Chapter 4 Preview. Slide 4-5. Chapter 4 Reading Quiz. Slide 4-6. Reading Question 4.1.

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Chapter 4 Lecture

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  1. Chapter 4 Lecture

  2. Chapter 4 Kinematics in Two Dimensions Chapter Goal: To learn how to solve problems about motion in a plane. Slide 4-2

  3. Chapter 4 Preview Slide 4-3

  4. Chapter 4 Preview Slide 4-4

  5. Chapter 4 Preview Slide 4-5

  6. Chapter 4 Reading Quiz Slide 4-6

  7. Reading Question 4.1 A ball is thrown upward at a 45 angle. In the absence of air resistance, the ball follows a • Tangential curve. • Sine curve. • Parabolic curve. • Linear curve. Slide 4-7

  8. Reading Question 4.1 A ball is thrown upward at a 45 angle. In the absence of air resistance, the ball follows a • Tangential curve. • Sine curve. • Parabolic curve. • Linear curve. Slide 4-8

  9. Reading Question 4.2 A hunter points his rifle directly at a coconut that he wishes to shoot off a tree. It so happens that the coconut falls from the tree at the exact instant the hunter pulls the trigger. Consequently, • The bullet passes above the coconut. • The bullet hits the coconut. • The bullet passes beneath the coconut. • This wasn’t discussed in Chapter 4. Slide 4-9

  10. Reading Question 4.2 A hunter points his rifle directly at a coconut that he wishes to shoot off a tree. It so happens that the coconut falls from the tree at the exact instant the hunter pulls the trigger. Consequently, • The bullet passes above the coconut. • The bullet hits the coconut. • The bullet passes beneath the coconut. • This wasn’t discussed in Chapter 4. Slide 4-10

  11. Reading Question 4.3 When discussing relative motion, the notation AB means • The absolute velocity. • The AB-component of the velocity. • The velocity of A relative to B. • The velocity of B relative to A. • The velocity of the object labeled AB. Slide 4-11

  12. Reading Question 4.3 When discussing relative motion, the notation AB means • The absolute velocity. • The AB-component of the velocity. • The velocity of A relative to B. • The velocity of B relative to A. • The velocity of the object labeled AB. Slide 4-11

  13. Reading Question 4.4 The quantity with the symbol ω is called • The circular weight. • The circular velocity. • The angular velocity. • The centripetal acceleration. • The angular acceleration. Slide 4-13

  14. Reading Question 4.4 The quantity with the symbol ω is called • The circular weight. • The circular velocity. • The angular velocity. • The centripetal acceleration. • The angular acceleration. Slide 4-14

  15. Reading Question 4.5 The quantity with the symbol  is called • The circular weight. • The circular velocity. • The angular velocity. • The centripetal acceleration. • The angular acceleration. Slide 4-15

  16. Reading Question 4.5 The quantity with the symbol  is called • The circular weight. • The circular velocity. • The angular velocity. • The centripetal acceleration. • The angular acceleration. Slide 4-16

  17. Reading Question 4.6 For uniform circular motion, the acceleration • Points toward the center of the circle. • Points away from the circle. • Is tangent to the circle. • Is zero. Slide 4-17

  18. Reading Question 4.6 For uniform circular motion, the acceleration • Points toward the center of the circle. • Points away from the circle. • Is tangent to the circle. • Is zero. Slide 4-18

  19. Chapter 4 Content, Examples, and QuickCheck Questions Slide 4-19

  20. Acceleration The average acceleration of a moving object is defined as the vector: The acceleration points in the same direction as , the change in velocity. As an object moves, its velocity vector can change in two possible ways: • The magnitude of the velocity can change, indicating a change in speed, or 2. The direction of the velocity can change, indicating that the object has changed direction. Slide 4-20

  21. Tactics: Finding the Acceleration Vector Slide 4-21

  22. Tactics: Finding the Acceleration Vector Slide 4-22

  23. QuickCheck 4.1 A particle undergoes acceleration while moving from point 1 to point 2. Which of the choices shows the velocity vector as the object moves away from point 2? Slide 4-23

  24. QuickCheck 4.1 A particle undergoes acceleration while moving from point 1 to point 2. Which of the choices shows the velocity vector as the object moves away from point 2? Slide 4-24

  25. Acceleration • The figure to the right shows a motion diagram of Maria riding a Ferris wheel. • Maria has constant speed but not constant velocity, so she is accelerating. • For every pair of adjacent velocity vectors, we can subtract them to find the average acceleration near that point. Slide 4-25

  26. Acceleration • At every point Maria’s acceleration points toward the center of the circle. • This is an acceleration due to changing direction, not to changing speed. Slide 4-26

  27. QuickCheck 4.2 A car is traveling around a curve at a steady 45 mph. Is the car accelerating? • Yes • No Slide 4-27

  28. QuickCheck 4.2 A car is traveling around a curve at a steady 45 mph. Is the car accelerating? • Yes • No Slide 4-28

  29. QuickCheck 4.3 A car is traveling around a curve at a steady 45 mph. Which vector shows the direction of the car’s acceleration? E. The acceleration is zero. Slide 4-29

  30. QuickCheck 4.3 A car is traveling around a curve at a steady 45 mph. Which vector shows the direction of the car’s acceleration? E. The acceleration is zero. Slide 4-30

  31. Example 4.1 Through the Valley Slide 4-31

  32. Example 4.1 Through the Valley Slide 4-32

  33. Analyzing the Acceleration Vector • An object’s acceleration canbe decomposed into components parallel and perpendicular to the velocity. • is the piece of the acceleration that causes the object to change speed. • is the piece of the acceleration that causes the object to change direction. • An object changing direction always has a component of acceleration perpendicular to the direction of motion. Slide 4-33

  34. QuickCheck 4.4 A car is slowing down as it drives over a circular hill. Which of these is the acceleration vector at the highest point? Slide 4-34

  35. Acceleration of changing speed Acceleration of changing direction QuickCheck 4.4 A car is slowing down as it drives over a circular hill. Which of these is the acceleration vector at the highest point? Slide 4-35

  36. Two-Dimensional Kinematics • The figure to the right shows the trajectory of a particle moving in the x-y plane. • The particle moves from position at time t1 to position at a later time t2. • The average velocity points in the direction of the displacement and is Slide 4-36

  37. Two-Dimensional Kinematics • The instantaneous velocity is the limit of avg as ∆t 0. • As shown the instantaneous velocity vector is tangent to the trajectory. • Mathematically: which can be written: where: Slide 4-37

  38. Two-Dimensional Kinematics • If the velocity vector’s angle  is measured from the positive x-direction, the velocity components are: where the particle’s speed is • Conversely, if we know the velocity components, we can determine the direction of motion: Slide 4-38

  39. Two-Dimensional Acceleration • The figure to the right shows the trajectory of a particle moving in the x-y plane. • The instantaneous velocity is at time t1and at a later time t2. • We can use vector subtraction to find avg during the time interval ∆tt2t1. Slide 4-39

  40. Two-Dimensional Acceleration • The instantaneous acceleration is the limit of avg as ∆t 0. • The instantaneous acceleration vector is shown along with the instantaneous velocity in the figure. • By definition, is the rate at which is changing at that instant. Slide 4-40

  41. Decomposing Two-Dimensional Acceleration • The figure to the right shows the trajectory of a particle moving in the x-y plane. • The acceleration is decomposed into components and . • is associated with a change in speed. • is associated with a change of direction. • always points toward the “inside” of the curve because that is the direction in which is changing. Slide 4-41

  42. Decomposing Two-Dimensional Acceleration • The figure to the right shows the trajectory of a particle moving in the x-y plane. • The acceleration is decomposed into components ax and ay. • If vx and vy are the x- and y- components of velocity, then Slide 4-42

  43. Constant Acceleration • If the acceleration is constant, then the two components ax and ay are both constant. • In this case, everything from Chapter 2 about constant-acceleration kinematics applies to the components. • The x-components and y-components of the motion can be treated independently. • They remain connected through the fact that ∆t must be the same for both. Slide 4-43

  44. Projectile Motion • Baseballs, tennis balls, Olympic divers, etc. all exhibit projectile motion. • A projectile is an object that moves in two dimensions under the influence of only gravity. • Projectile motion extends the idea of free-fall motion to include a horizontal component of velocity. • Air resistance is neglected. • Projectiles in two dimensions follow a parabolic trajectory as shown in the photo. Slide 4-44

  45. Projectile Motion • The start of a projectile’s motion is called the launch. • The angle  of the initial velocity v0above the x-axis is called the launch angle. • The initial velocity vector can be broken into components. where v0is the initial speed. Slide 4-45

  46. Projectile Motion • Gravity acts downward. • Therefore, a projectile has no horizontal acceleration. • Thus: • The vertical component of acceleration ay is g of free fall. • The horizontal component of ax is zero. • Projectiles are in free fall. Slide 4-46

  47. Projectile Motion • The figure shows a projectile launched from the origin with initial velocity: • The value of vx never changes because there’s no horizontal acceleration. • vydecreases by 9.8 m/s every second. Slide 4-47

  48. Example 4.4 Don’t Try This at Home! Slide 4-48

  49. Example 4.4 Don’t Try This at Home! Slide 4-49

  50. Example 4.4 Don’t Try This at Home! Slide 4-50

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