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Motion in a Plane: Velocity, Acceleration, and Projectile Motion

This chapter explores motion in a plane, including vector addition, graphical addition of vectors, velocity, acceleration, and projectile motion. Examples and problem-solving strategies are provided.

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Motion in a Plane: Velocity, Acceleration, and Projectile Motion

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  1. Chapter 3 Motion in a Plane Chapter 3 - Revised: 2/1/2010

  2. Motion in a Plane • Vector Addition • Velocity • Acceleration • Projectile motion Chapter 3 - Revised: 2/1/2010

  3. Graphical Addition and Subtraction of Vectors A vector is a quantity that has both a magnitude and a direction. Position is an example of a vector quantity. A scalar is a quantity with no direction. The mass of an object is an example of a scalar quantity. Chapter 3 - Revised: 2/1/2010

  4. Notation Vector: The magnitude of a vector: The direction of vector might be “35 south of east”; “20 above the +x-axis”; or…. Scalar: m (not bold face; no arrow) Chapter 3 - Revised: 2/1/2010

  5. F2 R F1 Graphical Addition of Vectors To add vectors graphically they must be placed “tip to tail”. The result (F1 + F2) points from the tail of the first vector to the tip of the second vector. This is sometimes called the resultant vector R Chapter 3 - Revised: 2/1/2010

  6. Vector Simulation Chapter 3 - Revised: 2/1/2010

  7. Examples • Trig Table • Vector Components • Unit Vectors Chapter 3 - Revised: 2/1/2010

  8. Graphical Subtraction of Vectors Think of vector subtraction A B as A+(B), where the vector B has the same magnitude as B but points in the opposite direction. Vectors may be moved any way you please (to place them tip to tail) provided that you do not change their length nor rotate them. Chapter 3 - Revised: 2/1/2010

  9. Velocity Chapter 3 - Revised: 2/1/2010

  10. The instantaneous velocity points tangent to the path. vi r vf ri rf Points in the direction of r A particle moves along the blue path as shown. At time t1 its position is ri and at time t2 its position is rf. y x Chapter 3 - Revised: 2/1/2010

  11. A displacement over an interval of time is a velocity The instantaneous velocity is represented by the slope of a line tangent to the curve on the graph of an object’s position versus time. Chapter 3 - Revised: 2/1/2010

  12. Acceleration Chapter 3 - Revised: 2/1/2010

  13. Points in the direction of v. v vf ri rf The instantaneous acceleration can point in any direction. A particle moves along the blue path as shown. At time t1 its position is r0 and at time t2 its position is rf. y vi x Chapter 3 - Revised: 2/1/2010

  14. A nonzero acceleration changes an object’s state of motion These have interpretations similar to vav and v. Chapter 3 - Revised: 2/1/2010

  15. Motion in a Plane with Constant Acceleration - Projectile What is the motion of a struck baseball? Once it leaves the bat (if air resistance is negligible) only the force of gravity acts on the baseball. Acceleration due to gravity has a constant value near the surface of the earth. We call it g = 9.8 m/s2 Only the vertical motion is affected by gravity Chapter 3 - Revised: 2/1/2010

  16. Projectile Motion The baseball has ax = 0 and ay = g, it moves with constantvelocity along the x-axis and with a changing velocity along the y-axis. Chapter 3 - Revised: 2/1/2010

  17. Example: An object is projected from the origin. The initial velocity components are vix = 7.07 m/s, and viy = 7.07 m/s. Determine the x and y position of the object at 0.2 second intervals for 1.4 seconds. Also plot the results. Since the object starts from the origin, y and x will represent the location of the object at time t. Chapter 3 - Revised: 2/1/2010

  18. Example continued: Chapter 3 - Revised: 2/1/2010

  19. Example continued: This is a plot of the x position (black points) and y position (red points) of the object as a function of time. Chapter 3 - Revised: 2/1/2010

  20. Example continued: This is a plot of the y position versus x position for the object (its trajectory). The object’s path is a parabola. Chapter 3 - Revised: 2/1/2010

  21. y vi 60° x Example (text problem 3.50): An arrow is shot into the air with  = 60° and vi = 20.0 m/s. (a) What are vx and vy of the arrow when t = 3 sec? The components of the initial velocity are: CONSTANT At t = 3 sec: Chapter 3 - Revised: 2/1/2010

  22. y r x Example continued: (b) What are the x and y components of the displacement of the arrow during the 3.0 sec interval? Chapter 3 - Revised: 2/1/2010

  23. Example: How far does the arrow in the previous example land from where it is released? The arrow lands when y = 0. Solving for t: What about the 2nd solution? The distance traveled is: Chapter 3 - Revised: 2/1/2010

  24. Summary • Adding and subtracting vectors (graphical method & component method) • Velocity • Acceleration • Projectile motion (here ax = 0 and ay = g) Chapter 3 - Revised: 2/1/2010

  25. Projectiles Examples • Problem solving strategy • Symmetry of the motion • Contact forces versus long-range forces • Dropped from a plane • The home run Chapter 3 - Revised: 2/1/2010

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