1. Chapter 3 Kinematics in Two Dimensions; Vectors 1

2. 3-1 Vectors and Scalars A vector has magnitude as well as direction. (displacement, velocity, force, momentum) A scalar has only a magnitude. (mass, time, temperature) 2

3. 3-2 Addition of Vectors – Graphical Methods For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates. 3

4. Addition of vectors that are vertical or horizontal only 4

5. If the motion is in two dimensions, the situation gets more complicated. 5

6. 2 Dimensional Kinematics We need to use vector diagrams to describe the motion 6

7. Describing the direction or angle of the vector 7

8. Describing the magnitude of the vector 8

9. 3-2 Addition of Vectors – Graphical Methods Even if the vectors are not at right angles, they can be added graphically using the “tail-to-tip” method. 9

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11. 3-2 Addition of Vectors – Graphical Methods If the paths are at right angles to one another; we can find the resultant by using the Pythagorean Theorem. 11

12. Using the Pythagorean Theorem to find the magnitude of the vector 12

13. 3-4 Adding Vectors by Components If the components are perpendicular, they can be found using trigonometric functions. Remember: SOH-CAH-TOA! 13

14. To determine the direction or angle of the resultant vector 14 Tan 11 1 Tan 1)

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16. Vertical vector components are noted with a y and horizontal with an x 16

17. 17 FxFx FyFy FxFx FxFx FxFx FyFy FyFy FyFy

18. 3-4 Adding Vectors by Components Adding vectors: 1. Draw a diagram. 2. Choose x and y axes. 3. Resolve each vector into x and y components. 4. Calculate each component using trig functions. 5. Add the components in each direction. 6. To find the magnitude and direction of the vector, use: 18

19. 3-5 Projectile Motion A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola. 19

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21. Can be understood by analyzing the horizontal and vertical motions separately. The x and y components are independent of each other! 21

22. Horizontal Motion (x) Vertical Motion (y) Forces (Present? - Yes or No. If present, what direction?) No Yes The force of gravity acts downward Acceleration (Present? - Yes or No. If present, what direction?) No Yes "g" is downward at 9.8 m/s/s Velocity (Constant or Changing?) Constant Changing (by 9.8 m/s each second) Analyzing parabolic motion 22

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24. At 1 sec time intervals 24

25. 3-5 Projectile Motion If an object is launched at an initial angle of θ 0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component. 25

26. Objects launched at an angle 26

27. Our previously learned equations still work! We just have to analyze the x and y components of the motion separately. 27

28. 3-6 Solving Problems Involving Projectile Motion 1. Read the problem carefully, and choose the object(s) you are going to analyze. 2. Draw a diagram. 3. Choose an origin and a coordinate system. 4. Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g. 5. Examine the x and y motions separately. 6. List known and unknown quantities. Remember that v x never changes, and that v y = 0 at the highest point. 7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them. 28

29. It’s all relative! 29

30. Magnitude of the resultant A 2 + B 2 = R 2 (100 km/hr) 2 + (25 km/hr) 2 = R 2 R= 103.1 km/hr Direction of resultant tan Ө = opp/adj tan Ө = (25/100) Ө = 14.0 degrees 30

31. Each velocity is labeled first with the object, and second with the reference frame in which it has this velocity. Therefore, v WS is the velocity of the water in the shore frame, v BS is the velocity of the boat in the shore frame, and v BW is the velocity of the boat in the water frame. 31

32. 3-8 Relative Velocity In this case, the relationship between the three velocities is: (3-6) 32

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34. References Giancoli, Douglas. Physics: Principles with Applications 6th Edition. 2009. http://www.physicsclassroom.com 34