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Motion Along Two or Three Dimensions

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Motion Along Two or Three Dimensions. Review. Equations for Motion Along One Dimension. Review. Motion Equations for Constant Acceleration. 1. 2. 3. 4. Slow Down. Giancoli Problem 3-9.

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Motion Along Two or Three Dimensions

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  1. Motion Along Two or Three Dimensions

  2. Review • Equations for Motion Along One Dimension

  3. Review • Motion Equations for Constant Acceleration • 1. • 2. • 3. • 4.

  4. Slow Down

  5. Giancoli Problem 3-9 • An airplane is traveling 735 km/hr in a direction 41.5o west of north. How far North and how far West has the plane traveled after 3 hours?

  6. Problem Solving Strategy • Define your origin • Define your axis • Write down the given (as well as what you’re looking for) • Reduce the two dimensional problem into two one dimensional problems. • Choose which of the four equations would work best

  7. Giancoli Problem 3-9 41.5

  8. Vector Addition

  9. Giancoli Problem 3-9 41.5

  10. Giancoli Problem 3-9 • In the Western direction • In the Northern direction 41.5

  11. Giancoli Problem 3-9 41.5

  12. X and Y components are independent • What happens along x does not affect y • What happens along y does not affect x • We can break down 2 dimensional motion as if we’re dealing with two separate one dimensional motions,

  13. Serway Problem 3-25 • While exploring a cave, a spelunker starts at the entrance and moves the following distances. She goes 75.0m N, 250m E, 125m at an angle 30.0 N of E, and 150m S. Find the resultant displacement from the cave entrance. • NOT DRAWN TO SCALE

  14. Serway Problem 3-25 • If you have the time and patience you can draw this system and solve the problem graphically. Or • Separate the vectors into their components. • NOT DRAWN TO SCALE

  15. Serway Problem 3-25 • NOT DRAWN TO SCALE

  16. Serway Problem 3-25 • Where • Substitute • NOT DRAWN TO SCALE

  17. Serway Problem 3-25 • CAUTION • NOT DRAWN TO SCALE

  18. Serway Problem 3-25 • NOT DRAWN TO SCALE

  19. NOT DONE YET 2 degrees S of E • NOT DRAWN TO SCALE

  20. Lets add another dimension • Serway 3-44 • A radar station locates a sinking ship at range 17.3 km bearing 136o clockwise from north. From the same station, a rescue plane is at horizontal range 19.6 km, 153o clockwise from north, with elevation 2.20 km. a) find position vector for the ship relative to the plane, letting i represent East, j represent north and k up. b) How far apart are the plane and the ship?

  21. Serway 3-44 Vectors • S = Radar to ship • P=Radar to plane • Vector of Plane to ship? • Let D be plane to ship • Then

  22. Express vectors in terms of their components

  23. Similarly

  24. Vector Addition (Subtraction)

  25. Magnitude of Vector D

  26. Car on a Curve

  27. What is the Velocity?

  28. Velocity on a Curve

  29. Lets make Δt smaller

  30. Lets make Δt smaller and smaller

  31. Velocity on a Curve • Velocity is tangent to the path

  32. Velocity on a Curve • We can find direction of • velocity at any point in time • Velocity is changing

  33. Acceleration on a Curve

  34. Acceleration on a Curve

  35. Acceleration on a Curve • Average Acceleration is changing • Acceleration is not constant

  36. Special Cases • We’re not yet equipped to deal with non-constant acceleration. • So lets first examine some situations where acceleration is constant.

  37. Projectile Motion • A projectile is any body that is given an initial velocity and then follows a path determined entirely by gravity and air resistance. • For simplicity lets ignore air resistance first. • The trajectory is the path a projectile takes. • We don’t care about how the projectile was launched or how it lands. We only care about the motion when it’s in free fall.

  38. Projectile Motion - Trajectory • Follows Parabolic path (proof algebra) • Velocity is always tangent to the path • Since acceleration is purely downwards, motion is constrained to two dimension.

  39. Projectile Motion - Trajectory

  40. Projectile Motion - Trajectory

  41. Projectile Motion - Components • Reduce the velocity vector to its components. • These components are orthogonal to each other so they have no effect on each other. • Motion along each axis is independent. • We can then use the equations of motion in one direction.

  42. Equations for Motion with constant Acceleration • x-axis • 1. • 2. • 3. • 4. • y-axis • 1. • 2. • 3. • 4.

  43. But Wait • In projectile motion, only gravity is acting on the object • a=-g=-9.80m/s2 • What are the components of this acceleration

  44. But Wait • In projectile motion, only gravity is acting on the object • a=-g=-9.80 m/s2 • What are the components of this acceleration • ay=-9.80 m/s2 • ax= 0 there is NO x-component

  45. Equations of Motion for Projectile Motion • x-axis (ax=0) • 1. • 2. • 3. • 4. • y-axis (ay=-g) • 1. • 2. • 3. • 4.

  46. Equations of Motion for Projectile Motion • x-axis (ax=0) • 1. • 2. • y-axis (ay=-g) • 1. • 2. • 3. • 4.

  47. Example • A motorcycle stuntman rides over a cliff. Just at the cliff edge his velocity is completely horizontal with magnitude 9.0 m/s. Find the motorcycles position, distance from the cliff edge, and velocity after 0.50s.

  48. List the given • Origin is cliff edge • a=-g=-9.80m/s2 • At time t=0s • At time t=0.50s

  49. Split into components

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