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Motion Along Two or Three Dimensions

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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