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Ch 6 Vectors

Ch 6 Vectors. 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?. Vector vs. Scalar.

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Ch 6 Vectors

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  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 Distance Velocity Acceleration Vector Vector Scalar Vector Displacement Speed Temperature Force Vector Vector Scalar Scalar Pressure Energy Volume Vector Scalar Scalar

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

  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 head 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. to tip of last vector Drawn from tail of first vector

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

  8. 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 Resultant

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

  10. Adding Vectors Graphically • When graphically adding vectors: • The scale must not change • The direction of the reference angle must not change

  11. 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

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

  13. Adding Vectors Graphically • Multimedia • Vector addition, order does not mater.

  14. 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.

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

  16. 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

  17. H Relative Velocity

  18. Vx Vy V Dx H Dy D Relative Velocity

  19. Vy Dy H Dx D V Vx Relative Velocity

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

  21. Independence of Vectors Dx Vx Dy D Vy V A boat travels north at 8m/s across an 80m wide river which flows west at 5m/s . The river is 80m wide

  22. Independence of Vectors • Perpendicular vector quantities are independent of each other. • For example in projectile motion • Vx Velocity in the X-direction • Vy Velocity in the Y-direction Are independent of each other.

  23. Trig functions take angles for input and give ratios for their output. Inverse Trig functions take ratios for input and give angles for output. Trig Functions Inverse Trig Functions Trig Function Reminders Angle Ratio Ratio Angle

  24. hypotenuse opposite adjacent Adding Force Vectors Analytically

  25. 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.

  26. -5 5 -8.66 8.66 8.66 -5 5 -8.66

  27. Independence of Vectors Dx Vx Dy D Vy V 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 At what angle did the boat go?

  28. Vx Vy V Example solution • What is the resultant velocity of the boat

  29. Example solution • How much time does it take to cross the river? • How far down stream does the boat go.

  30. Adding Vectors Analytically

  31. Adding Vectors Analytically Add the y components together Add the x components together Compute the Resultant

  32. 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

  33. Adding Vectors Analytically

  34. Adding Vectors WS 17 • Analytically and Graphically add the following vector sets. • v1 17m/s @ 300 • v2 24m/s @ 170 • v3 24m/s @ 55o • v4 19m/s @ 20o

  35. Practice Problem WS6a #1

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

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

  38. 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?

  39. Vector Concept questions • A vector is to be added graphically, which, if any, of the following may you do the first vector? • Rotate it • Move it • Lengthen it • Shorten it

  40. 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?

  41. Adding Vectors • Graphically and Analytically add the following vector sets. • V1 5.2m/s @ 70 • V2 6.4m/s @ 210 • V1 10m/s @ 45 • V2 15m/s @ 135

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

  43. 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 ?

  44. Blocks are 1 cm x 1 cm Scale: 1 cm = 2 miles Quiz Solution

  45. End Ch6 Vectors

  46. Two Body Probems

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

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