Understanding Motion in a Plane with Vectors

1. Chapter 3 Motion in a Plane Chapter 3b - Revised: 6/7/2010

2. Motion in a Plane • Vector Addition • Velocity • Acceleration • Projectile motion Chapter 3b - Revised: 6/7/2010

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 3b - Revised: 6/7/2010

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 3b - Revised: 6/7/2010

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 3b - Revised: 6/7/2010

6. Vector Simulation Chapter 3b - Revised: 6/7/2010

7. Examples • Trig Table • Vector Components • Unit Vectors Chapter 3b - Revised: 6/7/2010

8. Types of Vectors Chapter 3b - Revised: 6/7/2010

9. Relative Displacement Vectors Vector Addition Vector Subtraction is a relative displacement vector of point P3 relative to P2 Chapter 3b - Revised: 6/7/2010

10. Vector Addition via Parallelogram Chapter 3b - Revised: 6/7/2010

11. Graphical Method of Vector Addition Chapter 3b - Revised: 6/7/2010

12. 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 3b - Revised: 6/7/2010

13. Vector Components Chapter 3b - Revised: 6/7/2010

14. Vector Components Chapter 3b - Revised: 6/7/2010

15. Graphical Method of Vector Addition Chapter 3b - Revised: 6/7/2010

16. Unit Vectors in Rectangular Coordinates Chapter 3b - Revised: 6/7/2010

17. Vector Components in Rectangular Coordinates Chapter 3b - Revised: 6/7/2010

18. Vectors with Rectangular Unit Vectors Chapter 3b - Revised: 6/7/2010

19. Dot Product - Scalar The dot product multiplies the portion of A that is parallel to B with B Chapter 3b - Revised: 6/7/2010

20. Dot Product - Scalar In 2 dimensions In any number of dimensions The dot product multiplies the portion of A that is parallel to B with B Chapter 3b - Revised: 6/7/2010

21. Cross Product - Vector The cross product multpilies the portion of A that is perpendicular to B with B Chapter 3b - Revised: 6/7/2010

22. Cross Product - Vector In 2 dimensions In any number of dimensions Chapter 3b - Revised: 6/7/2010

23. Velocity Chapter 3b - Revised: 6/7/2010

24. The instantaneous velocity points tangent to the path. vi r vf ri rf Points in the direction of r A particle moves along the curved path as shown. At time t1 its position is ri and at time t2 its position is rf. y x Chapter 3b - Revised: 6/7/2010

25. 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 3b - Revised: 6/7/2010

26. Acceleration Chapter 3b - Revised: 6/7/2010

27. Points in the direction of v. v vf ri rf A particle moves along the curved path as shown. At time t1 its position is r0 and at time t2 its position is rf. y vi x Chapter 3b - Revised: 6/7/2010

28. A nonzero acceleration changes an object’s state of motion These have interpretations similar to vav and v. Chapter 3b - Revised: 6/7/2010

29. 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 3b - Revised: 6/7/2010

30. 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 3b - Revised: 6/7/2010

31. 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 3b - Revised: 6/7/2010

32. Example continued: Chapter 3b - Revised: 6/7/2010

33. 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 3b - Revised: 6/7/2010

34. 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 3b - Revised: 6/7/2010

35. 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 3b - Revised: 6/7/2010

36. 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 3b - Revised: 6/7/2010

37. 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: Chapter 3b - Revised: 6/7/2010

38. Summary • Adding and subtracting vectors (graphical method & component method) • Velocity • Acceleration • Projectile motion (here ax = 0 and ay = g) Chapter 3b - Revised: 6/7/2010

39. Projectiles Examples • Problem solving strategy • Symmetry of the motion • Dropped from a plane • The home run Chapter 3b - Revised: 6/7/2010