1. Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function of displacement (D)By computing the time to bottom, then computing the velocity.

3. Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function of displacement (D)By computing the time to bottom, then computing the velocity.

4. Chapter 12-5 Curvilinear Motion X-Y Coordinates

6. Here is the solution in Mathcad

7. Example: Hit target at Position (360’, -80’)

8. Example: Hit target at Position (360, -80)

9. 12.7 Normal and Tangential Coordinates u t : unit tangent to the path u n : unit normal to the path

10. Normal and Tangential Coordinates Velocity Page 53

11. Normal and Tangential Coordinates ‘e’ denotes unit vector (‘u’ in Hibbeler)

12. ‘e’ denotes unit vector (‘u’ in Hibbeler)

13. 12.8 Polar coordinates

14. Polar coordinates ‘e’ denotes unit vector (‘u’ in Hibbeler)

15. Polar coordinates ‘e’ denotes unit vector (‘u’ in Hibbeler)

16. 12.8 Polar coordinates In a polar coordinate system, the velocity vector can be written as v = v r u r + v θ u θ = ru r +r  u . The term  is called A) transverse velocity. B) radial velocity. C) angular velocity. D) angular acceleration...

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20. 12.10 Relative (Constrained) Motion v A is given as shown. Find v B Approach: Use rel. Velocity: v B = v A +v B/A (transl. + rot.)

21. Vectors and Geometry x y  t  r(t) 

22. Given: vectors A and B as shown. The RESULT vector is: (A) RESULT = A - B (B) RESULT = A + B (C) None of the above

23. Given: vectors A and B as shown. The RESULT vector is: (A) RESULT = A - B (B) RESULT = A + B (C) None of the above

24. Make a sketch: A V_rel v_Truck B The rel. velocity is: V_Car/Truck = v_Car -vTruck 12.10 Relative (Constrained) Motion V_truck = 60 V_car = 65

25. Make a sketch: A V_river v_boat B The velocity is: (A)V_total = v+boat – v_river (B)V_total = v+boat + v_river 12.10 Relative (Constrained) Motion

26. Make a sketch: A V_river v_boat B The velocity is: (A)V_total = v+boat – v_river (B)V_total = v+boat + v_river 12.10 Relative (Constrained) Motion

27. Rel. Velocity example: Solution

28. Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind (blue vector) We solve Graphically (Vector Addition)

29. Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind An observer on land (fixed Cartesian Reference) sees V wind and v Boat. Land

30. Plane Vector Addition is two-dimensional. 12.10 Relative (Constrained) Motion vBvB vAvA v B/A

31. Example cont’d: Sailboat tacking against Northern Wind 2. Vector equation (1 scalar eqn. each in i- and j- direction). Solve using the given data (Vector Lengths and orientations) and Trigonometry 50 0 15 0 i

32. Chapter 12.10 Relative Motion

33. Vector Addition

34. Differentiating gives:

36. Exam 1 We will focus on Conceptual Solutions. Numbers are secondary. Train the General Method Topics: All covered sections of Chapter 12 Practice: Train yourself to solve all Problems in Chapter 12

37. Exam 1 Preparation: Start now! Cramming won’t work. Questions: Discuss with your peers. Ask me. The exam will MEASURE your knowledge and give you objective feedback.

38. Exam 1 Preparation: Practice: Step 1: Describe Problem Mathematically Step2: Calculus and Algebraic Equation Solving