1. Ch 6 Vectors

2. Vectors What is the difference between a scalar and a vector? A vector is a physical quantity that has both magnitude and direction What are some examples of vectors that we have used in this class?

3. Vector vs. Scalar State whether each of the following quantities is a vector or a scalar: Position AccelerationVelocity Speed Displacement Distance Force Energy Temperature Volume Vector Scalar Vector Pressure Vector

4. Representing Vectors Remember that vectors have magnitude AND direction. 0° 90° 180° 135° 45° 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

5. Adding Vectors Graphically Pick a scale for your drawing. Draw the first vector starting at the origin. Place your protractor at the “Head” of the first vector to make the correct angle. Draw the next vector such that it starts at the tip of the first vector. Continue to line each vector up head to tail. Draw the resultant vector. Measure its length for the magnitude and angle for its direction.

6. Resultant Vector Resultant Vector is the sum of 2 or more vectors. Drawn with a dashed line. Drawn from tail of first vector to tip of last vector

7. Expressing Vectors A golf ball is struck at an angle of  = 36° with the horizontal at a velocity of 45m/s. Strike v 1x v 1y 

8. Vectors can be added graphically by placing the “Tail” of one vector to the “Head” of the other.

9. The Resultant is the sum of components of two or more vectors –The resultant can be found by drawing a vector from the origin to the head of the last vector

10. If you walked 6 blocks East and then 4 blocks north What is your displacement? Adding Vectors Graphically

11. To find the magnitude of the resultant, measure the length To find the direction of the resultant, measure the angle. –The direction is always measured counter clockwise from the horizontal (east) Multimedia vector directionvector direction

12. Vector 1: 7cm East Vector 2: 5cm North Vector 3: 6cm West 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Adding Vectors Graphically

13. When graphically adding vectors: –The scale must not change –The direction of the reference angle must not change

14. Adding Vectors Graphically Does the order in which you add the vectors matter? 1+2+3=6 3+2+1=6 2+1+3=6

15. Adding Vectors Graphically You walk 5m @ 0 o and then turns to walk 6m @90 o. Finally, you turn to walk 8 m at 200°. What is your displacement? Addition is commutative!

16. Adding Vectors Graphically Vectors are always added head to tail Always measure the angle from the +x axis. Vectors are express in two parts Given vectors A and B, find vector C.

17. Adding Vectors Graphically Given vectors A and B, find vector C. C = B + A

18. Adding Vectors Graphically Multimedia –Vector addition, order does not mater.Vector addition, order does not mater.

19. Adding Vectors Graphically Vector 1: 300.0 m @ 0  Vector 2 450.0 m @ 135  Vector 3 250.0 m @ 270  What is the Resultant? D = A + B + C

20. Independence of Vectors Perpendicular vector quantities are independent of each other. For example in projectile motion –V x Velocity in the X-direction –V y Velocity in the Y-direction Are independent of each other.

21. Relative Velocity

24. Independence of Vectors A boat travels north at 8m/s across an 80m wide river which flows west at 5m/s. The river is 80m wide

25. Independence of Vectors A boat travels east at 8m/s across an 80m wide river which flows north at 5m/s. The river is 80m wide. a) How long does it take to cross the river?

26. Independence of Vectors A boat travels north at 8m/s across an 80m wide river which flows west at 5m/s. The river is 80m wide. b) How far west has the boat traveled?

27. Independence of Vectors Multimedia The river boat The plane and the wind

28. Independence of Vectors A boat head directly across a river 41m wide at 3.8m/s. The current is flowing downstream at 2.2m/s. What is the resultant velocity of the boat How much time does it take to cross the river? How far down stream does the boat go.

29. Adding Force Vectors Analytically hypotenuse adjacent opposite

30. Components of Vectors Finding the vector magnitude and direction when you know the components. Recall:  is measured from the positive x axis. Caution: Beware of the tangent function. Always consider in which quadrant the vector lies when dealing with the tangent function.

31. 8.66 5 -8.66 5 -5 8.66 -5

32. Independence of Vectors A boat head directly across a river 41m wide at 3.8m/s. The current is flowing downstream at 2.2m/s. What is the resultant velocity of the boat How much time does it take to cross the river? How far down stream does the boat go.

33. Example solution a)What is the resultant velocity of the boat

34. Example solution b)How much time does it take to cross the river? c)How far down stream does the boat go.

35. Adding Vectors Analytically Resolve each vector into its horizontal and vertical components Add all of the vertical components together Add all of the horizontal components together Draw a right triangle using the horizontal and vertical resultants

36. Adding Vectors Analytically AyAy AxAx =45 o A=7N B=8 C=6 Add the x components together Add the y components together Compute the Resultant

37. Adding Vectors Analytically MagnitudeAngleXY 7N45 o 4.95N 8N180 o -8N0N 6N270 o 0N-6N ---------------------------R x =-3.05NR y =-1.05N R=3.22NAngle =19 o +180 o = 199 o

38. Adding Force Vectors Graphically Add the following 3 vectors

39. Adding Vectors Analytically

40. Add the x components together Add the y components together Compute the Resultant

41. Adding Vectors Analytically MagnitudeAngleX componentY component 4.5N30 o 3.89N2.25N 7N210 o -6.06N-3.5N 6N150 o -5.19N3.0N ---------------------------R x =-7.36NR y =1.7N R=7.55NAngle =-13 o +180 o = 167 o

42. Analytically and Graphically add the following vector sets. v117m/s @ 300  v224m/s @ 170  v324m/s @ 55 o v419m/s @ 20 o Adding Vectors WS 17

43. Practice Problem WS6a #1 R=22.7m/sAngle = 43.4 o MagnitudeAngleRxRy 173008.5-14.7 24170-23.64.17 245513.7619.65 192017.856.49 --------------------------16.5115.62

44. Multiplying a Vector by a Scalar A A C = -1/2 A B = 2A C B A ½ A

45. Adding “-” Vectors C = A + B D = A - B D = A + (- B) A B C -B D  Add “negative” vectors by keeping the same magnitude but adding 180 degrees to the direction of the original vector.

46. Vector Concept questions What method is used to add vectors graphically? How is the resultant vector affected if the force vectors are added in a different order? What is equilibrium?

47. Vector Concept questions A vector is to be added graphically, which, if any, of the following may you do the first vector? a)Rotate it b)Move it c)Lengthen it d)Shorten it

48. Vector Concept questions What is the sum of three vectors that form a triangle? If these vectors are forces, what does the imply about the object the forces are acting on?

49. Adding Vectors Graphically and Analytically add the following vector sets. –V15.2m/s @ 70  –V26.4m/s @ 210  –V110m/s @ 45  –V215m/s @ 135 

50. Components of Vectors Vector resolution is the process of finding the two component vectors.

51. Graphical Vector Quiz On the first part of his flight, Jason flies his plane 5.0 miles due east ( = 5.0 miles @ 0  ). He then turns and flies 10.0 miles North West ( 10.0 miles @ 135  ). Finally, he turns due south and flies 3.0 miles ( 3.0 miles @ 270  ). What is his displacement from his takeoff point ?

52. Quiz Solution Blocks are 1 cm x 1 cmScale: 1 cm = 2 miles

53. End Ch6 Vectors

54. Two Body Probems

58. Adding Vectors Graphically Protractor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20