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Solving Motion Problems in a Plane - Chapter 4

Learn how to solve motion problems in a two-dimensional plane with examples and tactics for finding acceleration vectors. The chapter covers concepts like angular velocity, uniform circular motion, and relative motion. Test your knowledge with quick check questions provided within the content.

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Solving Motion Problems in a Plane - Chapter 4

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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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