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Chapter 4

Chapter 4. Motion in 2 Dimensions. Overview. The focus of this chapter is kinematics in 2-D Projectile Motion Uniform Circular Motion Tangential/Radial Acceleration Relative Motion. 4.1 Pos, Vel , Accel Vectors.

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Chapter 4

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  1. Chapter 4 Motion in 2 Dimensions

  2. Overview • The focus of this chapter is kinematics in 2-D • Projectile Motion • Uniform Circular Motion • Tangential/Radial Acceleration • Relative Motion

  3. 4.1 Pos, Vel, Accel Vectors • Extending what we know about 1-D (straight line) motion to 2-D (motion in xy plane) • r - position vector (x i + y j ) (Points from the origin) • Δr = rf - ri displacement vector (vector subtraction = tail to tail)

  4. 4.1 • - Average Velocity Vector (points along direction of Δr, t is a scalar) • - Instantaneous Velocity • First Derivative of the Position Vector Function with respect to time

  5. 4.1

  6. 4.1 • Acceleration- rate of change of velocity • - average acceleration • - instantaneous acceleration • First Derivative of the Velocity Vector Function • Second Derivative the Position Vector Function

  7. 4.1 • Remember- acceleration = rate of change of v Accel can be cause by changes in Magnitude (speed) Direction • Quick Quizzes Pg 80

  8. 4.2 2-D Motion with cons. Accel • We can study an object moving in two dimensions if its position vector as a function of time is know. • - Position function • - Velocity function

  9. 4.2 • Example 4.1 Pg 82

  10. 4.3 Projectile Motion • Projectile Motion • Easily studied with two assumptions. • Vertical Motion is equivalent to free fall (-g) • Air Resistance is Negligible • The path of the projectile (trajectory) is parabolic in shape. • Track the motion as two separate functions • Up and Down (free fall) • Left and Right (uniform motion)

  11. 4.3

  12. 4.3 • Quick Quizzes Pg 85 • Example 4.2 Pg 85 • Vertical Height (See Board Derivation) • Horizontal Range (see Board Derivation) • Maximum Range (45o)

  13. 4.3 • Example Problems 4.4-4.7

  14. 4.4 Uniform Circular Motion • An object following a circular path at constant speed. • Acceleration is due to changing direcition of the tangent velocity vector.

  15. 4.4 • Centripetal Acceleration • Points to the Center of the Circular Path • Perpendicular to the tangent velocity Quick Quizzes/Example Pg 93

  16. 4.5 Tangential and Radial Accel • If the speed of an object is not constant around a circular path • The portion of the acceleration due to changing direction- radial acceleration • The portion of the acceleration due to changing speed- tangential acceleration

  17. 4.5

  18. 4.5 • Tang. Accel • Radial Accel • Total Accel (Magnitude) • Total Unit Vector Accel

  19. 4.5

  20. 4.5 • Remember- for uniform circular motion at = 0 Quick Quizzes/Example Pg 95

  21. 4.6 Relative Velocity and Accel • How motion is observed from a moving frame of reference rather than fixed frame. • Airport People Movers (moving sidewalk) Fig 4.21 • Ball and Skateboard Fig 4.22

  22. 4.6 • The observed displacements and velocities are different to the two observers • The accelerations however remain the same assuming that the moving frame has constant speed.

  23. 4.6 Quick Quiz Pg 98 Examples 4.10, 4.11 Pg 98-99

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