1. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20101 Motion in a Plane Chapter 3

2. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20102 Motion in a Plane Vector Addition Velocity Acceleration Projectile motion

3. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20103 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. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20104 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. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20105 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. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20106 Vector Simulation

7. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20107 Examples Trig Table Vector Components Unit Vectors

8. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20108 Types of Vectors

9. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/20109 Relative Displacement Vectors Vector Addition Vector Subtraction is a relative displacement vector of point P 3 relative to P 2

10. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201010 Vector Addition via Parallelogram

11. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201011 Graphical Method of Vector Addition

12. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201012 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.

13. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201013 Vector Components

14. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201014 Vector Components

15. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201015 Graphical Method of Vector Addition

16. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201016 Unit Vectors in Rectangular Coordinates

17. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201017 Vector Components in Rectangular Coordinates

18. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201018 Vectors with Rectangular Unit Vectors

19. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201019 Dot Product - Scalar The dot product multiplies the portion of A that is parallel to B with B

20. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201020 Dot Product - Scalar The dot product multiplies the portion of A that is parallel to B with B In 2 dimensions In any number of dimensions

21. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201021 Cross Product - Vector The cross product multpilies the portion of A that is perpendicular to B with B

22. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201022 In 2 dimensions In any number of dimensions Cross Product - Vector

23. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201023 Velocity

24. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201024 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 curved path as shown. At time t 1 its position is r i and at time t 2 its position is r f.

25. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201025 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

26. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201026 Acceleration

27. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201027 y x vivi riri rfrf vfvf A particle moves along the curved 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.

28. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201028 A nonzero acceleration changes an object’s state of motion These have interpretations similar to v av and v.

29. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201029 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

30. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201030 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

31. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201031 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.

32. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201032 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:

33. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201033 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:

34. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201034 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.

35. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201035 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

36. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201036 (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:

37. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201037 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:

38. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201038 Summary Adding and subtracting vectors (graphical method & component method) Velocity Acceleration Projectile motion (here a x = 0 and a y =  g)

39. MFMcGraw-PHY 1401Chapter 3b - Revised: 6/7/201039 Projectiles Examples Problem solving strategy Symmetry of the motion Dropped from a plane The home run