1. MFMcGrawChapter 3 - Revised: 2/1/20101 Motion in a Plane Chapter 3

2. MFMcGrawChapter 3 - Revised: 2/1/20102 Motion in a Plane Vector Addition Velocity Acceleration Projectile motion

3. MFMcGrawChapter 3 - Revised: 2/1/20103 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.

4. MFMcGrawChapter 3 - Revised: 2/1/20104 Notation Vector: The magnitude of a vector: Scalar: m (not bold face; no arrow) The direction of vector might be “35  south of east”; “20  above the +x-axis”; or….

5. MFMcGrawChapter 3 - Revised: 2/1/20105 To add vectors graphically they must be placed “tip to tail”. The result (F 1 + F 2 ) points from the tail of the first vector to the tip of the second vector. This is sometimes called the resultant vector R F1F1 F2F2 R Graphical Addition of Vectors

6. MFMcGrawChapter 3 - Revised: 2/1/20106 Vector Simulation

7. MFMcGrawChapter 3 - Revised: 2/1/20107 Examples Trig Table Vector Components Unit Vectors

8. MFMcGrawChapter 3 - Revised: 2/1/20108 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. Graphical Subtraction of Vectors 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.

9. MFMcGrawChapter 3 - Revised: 2/1/20109 Velocity

10. MFMcGrawChapter 3 - Revised: 2/1/201010 y x riri rfrf Points in the direction of  r rr vivi The instantaneous velocity points tangent to the path. vfvf A particle moves along the blue path as shown. At time t 1 its position is r i and at time t 2 its position is r f.

11. MFMcGrawChapter 3 - Revised: 2/1/201011 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. A displacement over an interval of time is a velocity

12. MFMcGrawChapter 3 - Revised: 2/1/201012 Acceleration

13. MFMcGrawChapter 3 - Revised: 2/1/201013 y x vivi riri rfrf vfvf A particle moves along the blue path as shown. At time t 1 its position is r 0 and at time t 2 its position is r f. vv Points in the direction of  v. The instantaneous acceleration can point in any direction.

14. MFMcGrawChapter 3 - Revised: 2/1/201014 A nonzero acceleration changes an object’s state of motion These have interpretations similar to v av and v.

15. MFMcGrawChapter 3 - Revised: 2/1/201015 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/s 2 Only the vertical motion is affected by gravity

16. MFMcGrawChapter 3 - Revised: 2/1/201016 The baseball has a x = 0 and a y =  g, it moves with constant velocity along the x-axis and with a changing velocity along the y- axis. Projectile Motion

17. MFMcGrawChapter 3 - Revised: 2/1/201017 Example: An object is projected from the origin. The initial velocity components are v ix = 7.07 m/s, and v iy = 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.

18. MFMcGrawChapter 3 - Revised: 2/1/201018 t (sec)x (meters)y (meters) 000 0.21.411.22 0.42.832.04 0.64.242.48 0.85.662.52 1.07.072.17 1.28.481.43 1.49.890.29 Example continued:

19. MFMcGrawChapter 3 - Revised: 2/1/201019 This is a plot of the x position (black points) and y position (red points) of the object as a function of time. Example continued:

20. MFMcGrawChapter 3 - Revised: 2/1/201020 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.

21. MFMcGrawChapter 3 - Revised: 2/1/201021 Example (text problem 3.50): An arrow is shot into the air with  = 60° and v i = 20.0 m/s. (a) What are v x and v y of the arrow when t = 3 sec? The components of the initial velocity are: At t = 3 sec: x y 60° vivi CONSTANT

22. MFMcGrawChapter 3 - Revised: 2/1/201022 (b) What are the x and y components of the displacement of the arrow during the 3.0 sec interval? y x r Example continued:

23. MFMcGrawChapter 3 - Revised: 2/1/201023 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: The distance traveled is: What about the 2nd solution?

24. MFMcGrawChapter 3 - Revised: 2/1/201024 Summary Adding and subtracting vectors (graphical method & component method) Velocity Acceleration Projectile motion (here a x = 0 and a y =  g)

25. MFMcGrawChapter 3 - Revised: 2/1/201025 Projectiles Examples Problem solving strategy Symmetry of the motion Contact forces versus long-range forces Dropped from a plane The home run