2. Mechanics Lecture 2, Slide 1 Classical Mechanics Lecture 2-Two-dimensional Kinematics Today's Concepts: a) Vectors b) Projectile motion c) Reference frames Reminder Lectures are posted online @ http://www.physics.utah.edu/~springer/phys1500/lectures.html

3. Projectile Motion Example -Trebuchet Mechanics Lecture 2, Slide 2 https://www.youtube.com/watch?v=JQLYHt-DM0Q How far did it go? How high did it go? What was its launch velocity? How high did it go?

4. Projectile Motion Mechanics Lecture 2, Slide 3 http://www.sciencebuddies.org/science-fair- projects/project_ideas/ApMech_p013.shtml http://www.yale.edu/ynhti/curriculum/units/2012/4/12.0 4.03.x.html TrebuchetCatapault  Projectile Motion

5. Homework 1 Mechanics Lecture 2, Slide 4 13 students with 100% 21 students > 90% Great job! Dropped? Could not find links? 90-100 80-90 70-8060-70 50-60 40-50 0-1010-20 See me! 216 South Physics Bldg MW 3-5PM 801-585-1390

6. Smartphysics links Mechanics Lecture 2, Slide 5 Link to earlier unit

7. Homework Frustration A. True B. False Mechanics Lecture 1, Slide 6 I experienced some degree of “frustration” with the Homework Problems

8. Homework Accomplishment A. True B. False Mechanics Lecture 1, Slide 7 I experienced a sense of “accomplishment” solving the Homework Problems

9. Mechanics Lecture 2, Slide 8 Need Vectors!!! to describe motion in 2 & 3 dimensions

10. Mechanics Lecture 2, Slide 9 Vectors and 2-d kinematics –Main Points

11. Mechanics Lecture 2, Slide 10 Vectors and 2-d kinematics –Main Points

12. Vectors and 2d-kinematics Important Equations Mechanics Lecture 2, Slide 11 Kinematics

13. Lecture Thoughts Mechanics Lecture 2, Slide 12

14. Mechanics Lecture 2, Slide 13 The object is the same no matter how we chose to describe it Vectors  Think of a vector as an arrow. (An object having both magnitude and direction) http://mathforum.org/~klotz/Vectors/VandP.1.html Vector : directed line segment

15. Mechanics Lecture 2, Slide 14 The object is the same no matter how we chose to describe it AxAx AyAy Vectors-Cartesian Coordinates/ComponentsCartesian  Magnitude (length) Direction  Cartesian Components Depends on coordinate system Independent of coordinate system

16. Mechanics Lecture 2, Slide 15 The object is the same no matter how we chose to describe it Vectors-Polar CoordinatesPolar  Magnitude (length) Direction  Polar Components Depends on coordinate system Independent of coordinate system

17. Mechanics Lecture 2, Slide 16 Vectors

18. Mechanics Lecture 2, Slide 17 A vector can be defined in 2 or 3 (or even more) dimensions: Vectors in 3D

19. Mechanics Lecture 2, Slide 18 The object is the same no matter how we chose to describe it Vectors Decompose vector along unit vectors!!!

20. Mechanics Lecture 2, Slide 19 Vector Addition Add  Tail to Head Add Components!!!

21. Clicker Question A+B A. B. C. D. E. Mechanics Lecture 2, Slide 20 Vectors and are shown to the right. Which of the following best describes + A B C D E

22. Clicker Question A-B A. B. C. D. E. Mechanics Lecture 2, Slide 21 Vectors and are shown to the right. Which of the following best describes  A B C D E

23. Mechanics Lecture 2, Slide 22 A B C D E Vectors and are shown to the right. Which of the following best describes + Clicker Question A+2B

24. Mechanics Lecture 2, Slide 23 Acceleration Vector

25. Mechanics Lecture 2, Slide 24 Acceleration Vector

26. Mechanics Lecture 2, Slide 25 Kinematics in 3D

27. Mechanics Lecture 2, Slide 26 Checkpoint 1

28. Mechanics Lecture 2, Slide 27 Horizontal Vertical Boring Projectile Motion

29. Mechanics Lecture 2, Slide 28 A flatbed railroad car is moving along a track at constant velocity. A passenger at the center of the car throws a ball straight up. Neglecting air resistance, where will the ball land? A) Forward of the center of the car B) At the center of the car C) Backward of the center of the car correct Ball and car start with same x position and x velocity, Since a = 0 they always have same x position. Train Demo Clicker Question v train car

30. Moving Rail Car A. B. C. Mechanics Lecture 2, Slide 29 A flatbed railroad car is moving along a track at constant velocity. A passenger at the center of the car throws a ball straight up. Neglecting air resistance, where will the ball land? A) Forward of the center of the car B) At the center of the car C) Backward of the center of the car correct Ball and car start with same x position and x velocity, Since a x = 0 they always have same x position. v train car

31. Mechanics Lecture 2, Slide 30 Time spend in the air depends on the maximum height Maximum height depends on the initial vertical velocity v train car

32. Monkey troubles A. B. C. Mechanics Lecture 2, Slide 31 You are a vet trying to shoot a tranquilizer dart into a monkey hanging from a branch in a distant tree. You know that the monkey is very nervous, and will let go of the branch and start to fall as soon as your gun goes off. In order to hit the monkey with the dart, where should you point the gun before shooting? A) Right at the monkey B) Below the monkey C) Above the monkey

33. Mechanics Lecture 2, Slide 32 Monkey x  x o Dart x  v o t Shooting the Monkey…

34. Mechanics Lecture 2, Slide 33 Shooting the Monkey… y = v oy t  1 / 2 g t 2 Still works even if you shoot upwards! y = y o  1 / 2 g t 2 Dart hits the monkey

35. Mechanics Lecture 2, Slide 34 Projectile Motion & Frames of Reference

36. Mechanics Lecture 2, Slide 35 Checkpoint 2 A) Enemy 1 B) Enemy 2 C) They are both hit at the same time 53% of you had incorrect answer… Let’s try again. Enemy 1Destroyer Enemy 2

37. Enemy 1Destroyer Enemy 2 Checkpoint 2 A. B. C. Mechanics Lecture 2, Slide 36 …Which enemy ship gets hit first? A) Enemy 1 B) Enemy 2 C) Same B) The height of the shell fired at ship 2 is less, so ship 2 gets hit first.

38. Mechanics Lecture 2, Slide 37 Enemy 1Destroyer Checkpoint 3 A destroyer fires two shells with different initial speeds at two different enemy ships. The shells follow the trajectories shown. Which enemy ship gets hit first? Enemy 2 A) Enemy 1 B) Enemy 2 C) They are both hit at the same time 26% of you had incorrect answer… Let’s try again.

39. Checkpoint 3 A. B. C. Mechanics Lecture 2, Slide 38 Enemy 1DestroyerEnemy 2 …Which enemy ship gets hit first? A) Enemy 1 B) Enemy 2 C) Same C) they both achieve the same height so they remain in the air the same amount of time

40. Range Mechanics Lecture 2, Slide 39

41. Range Mechanics Lecture 2, Slide 40 MAXIMUM range OCCURS AT 45 0

42. Trigonometric Identity for range equation Mechanics Lecture 2, Slide 41 http://mathworld.wolfram.com/Cosine.html http://mathworld.wolfram.com/Sine.html

43. Trigonometric Identities relating sum and products Mechanics Lecture 2, Slide 42 List of trigonometric identities

44. Question 2 Mechanics Lecture 2, Slide 43

45. Question 2 Mechanics Lecture 2, Slide 44

46. Mechanics Lecture 2, Slide 45 Field Goal Example A field goal kicker can kick the ball 30 m/s at an angle of 30 degrees w.r.t. the ground. If the crossbar of the goal post is 3m off the ground, from how far away can he kick a field goal? y-direction v oy = v o sin(30 o ) = 15 m/s y = y o + v oy t + ½ at 2 3 m = 0 m + (15 m/s) t – ½ (9.8 m/s 2 ) t 2 t = 2.8 s or t = 0.22 s. x-direction v ox = v o cos(30 o ) = 26 m/s D = x o + v ox t + ½ at 2 = 0 m + (26 m/s)(2.8 s) + 0 m/s 2 (2.8 s ) 2 = 72.8 m D 3 m y x Illini Kicks 70 yard Field Goal

47. Mechanics Lecture 2, Slide 46 Vectors and 2d-kinematics – Main Points

48. Vectors and 2d-kinematics Important Equations Mechanics Lecture 2, Slide 47

49. Hyperphysics-Trajectories Mechanics Lecture 1, Slide 48 http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html

50. Hyperphysics-Trajectories Mechanics Lecture 1, Slide 49

51. Hyperphysics-Trajectories Mechanics Lecture 1, Slide 50

52. Pre-lecture 2 viewing times Mechanics Lecture 2, Slide 51