1. Motion Vectors

2. What is the difference between a vector and a scalar quantity?

3. Vectors  Quantities with magnitude, unit, and direction  What are some examples:  Displacement  Velocity  Acceleration  Force

4.  Scalar Quantity- has only a magnitude and a unit.  Example- 10 km/hr N is vector while 10km/hr is a scalar

5. A picture is worth a 1000 words.. Motion diagrams

6. Motion Diagrams  Arrows represent vectors  Connect the tail of one vector to the arrow tip of the other  Applies to all motion vectors

7. A person walks east on a moving sidewalk going east

8. A person walks east on a moving sidewalk going west

9. A bug crawls north then east

10. A car accelerates to the south then accelerates going to the west

11. Practice drawing motion diagrams  Motion Vector Diagram WS

12.  Do they only indicate direction?  No  They can indicate magnitude  They can be used to determine the overall displacement, velocity, etc.  Vectors are being “added”  This is called the Resultant Motion Diagrams can indicate more!

13.  Arrows represent vectors  Length of arrow corresponds to magnitude of vector  Connect the tail of one vector to the arrow tip of the other  No matter what route you take from point A to point B your final displacement vector will be the same  The final displacement vector is called the resultant vector  Draw in the resultant vector from the tail of the first vector to the arrow head of the second  Applies to all motion vectors Remember: Motion Diagrams

14. Practice adding vectors Ex 1  Draw the following to scale: Indicate scale and compass rose. Make sure you draw tail to tip for the two vectors. Indicate the resultant with a dotted line. The tip of the resultant meets the tip of the second vector. You walk 5 m to the north and then 8 m east Determine your resultant displacement graphically. What did you get? I got 9.7m.

15. Ex 1 8 m 5 m

16. Ex 1 Determine your resultant displacement graphically. What did you get? I got 9.7m.

17. Ex 1 “θ” or Theta, is any unknown angle but in this case it is the angle between the two vectors Use a protractor to determine the angle “θ” of your resultant. Place the protractor along the axis of the initial vector. Take the reading in reference to the resultant. Refer to diagram on next slide

18. Ex 1 θ 5 m 8 m

19. Ex 1 What value did you get with your protractor? I got 59° This would be described as 59° east of north since the resultant is east of the north axis.

20. WHICH ANGLE?  Describe angle in reference to vertical vector compared to horizontal vector: N of E  Describe angle in reference to horizontal vector compared to vertical vector: E of N  How are these 2 angles related?

21. Ex 1  Solving resultant mathmatically: How? Determine your resultant displacement using the Pythagorean formula. Did you get 9.43 m?

22. Ex 1 θ Adj 5m Opp 8m

23. Ex 1 Solve angle mathematically: How? Use SOH CAH TOA Label the sides in reference to θ. I would suggest using tan in case you measured the hypotenuse incorrectly. Using Tan -1 : Θ Tan -1 (8m/5m) = You should get 57.99° The direction is still East of North This is in the same range as what we got with the protractor.

24. Practice adding vectors Ex 2  Draw the following to scale: A car travels 4 m/s west then 7 m/s south Determine your resultant velocity graphically. What did you get? I got 8.2 m/s Determine your resultant displacement using the Pythagorean formula. Did you get 8.06 m/s?

25. Ex 2 4 m/s 7 m/s

26. Ex 2 Use a protractor to determine the angle “θ” of your resultant. Place the protractor along the axis of the initial vector. Take the reading in reference to the resultant. Refer to diagram on next slide

27. Ex 2 Adj 4 m/s Opp 7 m/s θ

28. Ex 2 What value did you get with your protractor? I got 60° This would be described as 60° south of west since the resultant is south of the west axis. How could you do this mathematically? Use tan Label the sides in reference to θ.

29. Ex 2 Using Tan -1 : Θ Tan -1 (7m/s /4m/s) = You should get 60.26° The direction is still South of West This is in the same range as what we got with the protractor.

30. Practice adding vectors Ex 3-4  Perform the following examples:  Ex 3 You walk 8 meters south, then 3 meters east.  Ex 4. You run at 8 m/s to the east, then 2 m/s to the south.  Ex 3  Displacement = 8.54 m at 20.56º E of S  Ex 4  Displacement = 8.24 m at 14.04º S of E

31. What are Components?  The sides that make the resultant vector …the legs

32. Ex 5 Components  You are given a resultant vector of 8 m/s at 30° N of E  We are going to work backwards!  Draw in the y component  Draw in the x component  How would you determine these? 30° 8 m/s

33. Ex 5 Components  y: opp = (sinθ)(hyp) opp = sin(30°)(8 m/s) y = 4 m/s  x: adj = (cosθ)(hyp) adj = (cos30°)(8m/s) x = 6.9 m/s  These are pretty close to our graphical values. 30° 8 m/s adj opp

34. Ex 6: You are given a resultant vector of 6.0 m/s at 10° N of W What is the northern component? What is the western component? Draw the diagram. Solve mathematically

35.  Northern: O= SH  O = sin(10º)(6.0 m/s) = 1.04 m/s North  Western: A = CH  A = cos(10º)(6.0 m/s) = 5.91 m/s West

36. Special Situations  NW, SW, NE, SE all indicate angles of?  45 °  Due North or due South indicates what angle?  90 °  Due East or due West indicates what angle? 0°0°

37. Sketch the following  3.5 m/s due South  8 m/s due East  6 m/s at 10° North of East

38. Why do Components? Ex. 7 Lets say you are on a hike and walk 3.5 m due south. You then turn and travel 8m due east. After resting you conitinue on for 6 m due north. What is your total distance? What is your resultant displacement in respect to the earth?

39. Ex. 7 Lets say you are on a hike and walk 3.5 m due south. You then turn and travel 8m due east. After resting you conitinue on for 6 m due north. What is your total distance? What is your resultant displacement in respect to the earth?

40. Ex. 7 What is your total distance? That is easy-just add up all the distances: 3.5 + 8m + 6m = 17.5m

41. Ex. 7 What is your resultant displacement in respect to the earth? Displacement is the shortest distance between start and finish:

42. Why do Components? Where is your start and finish? How would you determine displacement? Disp

43. Do you see the triangle? Can you determine the sides? How would you calculate the Displacement? What about direction? Disp

44. Make the Triangle. Determine the horizontal and vertical components. Determine resultant Displacement? What about direction? Disp

45. Quick Check How do you determine total distances? Add the components, do not consider direction (signs) Do not include the resultant When should I include angles and direction on problems? Anytime that you are working with a vector. Remember vectors have magnitude and direction When do you use the sin, cos, & tan keys When you have the angle and are looking for the ratio or a side When do you use the sin -1, cos -1, or tan -1 keys? When you have two sides and are looking for the angle

46. Quick Check How do you determine the resultant of two vectors at a right angle? Pythagoreans Theorem How do you determine the components of a resultant when given the magnitude and angle/direction of that resultant? Use cos(Θ)(H) for the adjacent side. Typically the X or horizontal axis Use sin(Θ)(H) for the opposite side. Typically the Y or vertical axis

48. Concept You are floating in a river. If you do not paddle, what determines how far downstream you go in 60 seconds? What if you were paddling straight across the river, how would this change?

49. Example 8  A girl scout elects to swim across the river. The river is 37.5 meters wide. A current flows downstream at a rate of 0.66 m/s. If she initially swims towards the boy scout camp (directly cross the river) at a rate of 1.73 m/s, how long will it take her to reach the far shore?

50.  Remember what the question asks.  How long does it take her to swim across?  To solve for time, what do we need to know?  Use velocity and displacement but only in reference to crossing the river. ] 37.5m 1.73 m/s

51.  v gs = d r /t  t = d r /v gs  t = 37.5m÷1.73m/s  t = 21.7 s ] 37.5m 1.73 m/s

52.  Where exactly does the girl scout end up on the far shore?  What do we need to know?  To determine displacement we need velocity and time but only in reference to downstream. ] 37.5m 1.73 m/s Example 8

53.  To find where she ends up, what is the downstream velocity?  0.66m/s  What is the time?  21.7 sec  Time is the same for both cross stream and downstream.  14.3 m downstream 0.66m/s

54.  When working with multiple vectors remember they are independent of one another although they have a net effect.  In the case of the girl scout, her overall (think resultant) velocity and direction changed.  Do you know how to solve for the apparent resultant velocity and direction?

55.  Resultant velocity? Make sure you only use velocity vectors!  c 2 = 1.73 2 + 0.66 2  c = 1.85 m/s  Which angle for direction?  θ Tan -1 = (0.66m/s / 1.73m/s)  θ = 20.88°  v r = 1.85 m/s at 20.88° downstream in respect to motion 0.66m/s 1.73 m/s c θ

56. OR

57.  Resultant velocity? Make sure you only use velocity vectors!  c 2 = 1.73 2 + 0.66 2  c = 1.85 m/s  Which angle for direction?  θ Tan -1 = (1.73m/s / 0.66m/s)  θ = 69.1°  v r = 1.85 m/s at 69.1° downstream in respect to shore 0.66m/s 1.73 m/s c θ

58. Resources