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Vectors (10)

Vectors (10). Modelling. B. r. = 6 i - 2 j. A. s = -4 i + 4 j. r. r. r. r. r. r. r. O. A. A. B. B. B. B. B. + s =. A. = + s. = ( 6 i - 2 j ) + ( -4 i + 4 j ). = 6 i - 2 j - 4 i + 4 j. r. = ( 2 i + 2 j ) m. B. Displacement.

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Vectors (10)

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  1. Vectors (10) Modelling

  2. B r = 6i - 2j A s = -4i + 4j r r r r r r r O A A B B B B B + s = A = + s = (6i - 2j) + (-4i + 4j) = 6i - 2j- 4i + 4j r = (2i + 2j) m B Displacement An object with the position vector (6i - 2j)m is displaced by (-4i + 4j)m, what is it’s final position vector?

  3. Velocity Vector Calculation (1) r r + s = r r A A B B s = - s A r = 5i + 5j If a ship has a position vector 5i + 5j, 3 seconds later it has the position vector 2i - 4j What is the average velocity in the time? A Firstly, what is the displacement s? O r B = 2i - 4j s = (2i - 4j) - (5i + 5j) B s = 2i - 4j - 5i - 5j s = -3i - 9j

  4. Velocity Vector Calculation (2) average velocity = change in position vector time taken = (-3i - 9j) m average velocity s 3 sec. = (-i - 3j) ms-1 average velocity A r = 5i + 5j If a ship has a position vector 5i + 5j, 3 seconds later it has the position vector 2i - 4j What is the average velocity in the time? A s = -3i - 9j O r B = 2i - 4j B Meaning 1 ms-1 West and 3 ms-1 South

  5. Magnitude of the Velocity = (-i - 3j) ms-1 average velocity Magnitude of velocity = (32 + 12) = 10 = 3.2 ms-1 s 3 1

  6. Resultant Velocity (1) Something like this If the boat sails straight across, which direction will it actually go? A River 5ms-1 2ms-1 Direction of flow

  7. Resultant Velocity (2) a 5ms-1 The resultant  2ms-1 5ms-1 Direction tan  = opp/adj = 2/5  = tan-1(0.4) = 21.8o 2ms-1 or a = 90 - 21.8 = 68.2o measured from the river bank The RESULTANT VELOCITY can be found by a triangle of velocities Resultant = 5i + 2j Magnitude = (52 + 22) = 29 = 5.4 ms-1

  8. Resultant Velocity (3) = 5i - 2j NE Wind = 3i + 3j A plane flys with velocity 5i - 2j The wind blows with velocity 3i + 3j What is the resultant velocity and direction? Do a sketch

  9. Resultant Velocity (3.5) - sketch resultant velocity v Wind 3i + 3j 5i - 2j Wind 5i - 2j 3i + 3j resultant velocity v A plane flys with velocity 5i - 2j The wind blows with velocity 3i + 3j What is the resultant velocity and direction? You can sketch it either way round. The resultant is identical in both cases.

  10. Resultant Velocity (4) resultant velocity v A plane flys with velocity 5i - 2j The wind blows with velocity 3i + 3j What is the resultant velocity and direction? Wind 5i - 2j 3i + 3j v = 3i + 3j +5i - 2j = 8i + j

  11. Resultant Velocity (5) N 1 b  8 Direction tan  = opp/adj = 1/8  = tan-1(1/8) = 7.1o Bearing b = 90 - 7.1 = 82.9o A plane flys with velocity 5i - 2j The wind blows with velocity 3i + 3j What is the resultant velocity and direction? resultant velocity v v= 8i + j Magnitude = (82 + 12) = 65 = 8.1 ms-1

  12. P 50km O 70km OP = 700i + 500j Modeling Techniques A plane is 700km East and 500km North of the origin (airport). It flies with velocity 300i - 220j km/h. What is it’s Position Vector at time t hours. 1st: Find in terms of i and j the original position vector of the plane.

  13. OP = 700i + 500j Modeling Techniques A plane is 700km East and 500km North of the origin (airport). It flies with velocity 300i - 220j km/h. What is it’s Position Vector at time t hours. P motion O

  14. OP = 700i + 500j PC = 300i - 220j OC = 700i + 500j + 300i - 220j OC = 1000i + 280j Modeling Techniques A plane is 700km East and 500km North of the origin (airport). It flies with velocity 300i - 220j km/h. What is it’s Position Vector at time t hours. P C O 1 hour later

  15. OP = 700i + 500j PC = 600i - 440j OC = 700i + 500j + 600i - 440j OC = 1300i + 60j Modeling Techniques A plane is 700km East and 500km North of the origin (airport). It flies with velocity 300i - 220j km/h. What is it’s Position Vector at time t hours. P C O 2 hours later

  16. OP = 700i + 500j PC = 5(300i - 440j) OC = 700i + 500j + 1500i - 2200j OC = 2200i - 700j Modeling Techniques A plane is 700km East and 500km North of the origin (airport). It flies with velocity 300i - 220j km/h. What is it’s Position Vector at time t hours. P O C 5 hours later

  17. OP = 700i + 500j PC = t(300i - 220)j OC = 700i + 500j + t(300i - 220j) Original Point Parallel Vector Modeling Techniques A plane is 700km East and 500km North of the origin (airport). It flies with velocity 300i - 220j km/h. What is it’s Position Vector at time t hours. P Distance = speed x time C O t hours later

  18. [ ] -3 4 F (0,0) O L (6,-6) OL = 6i - 6j Modeling A Problem: example - 1 An Ocean Liner is at (6, -6) is cruising at 10 km/h in the direction . A fishing boat is anchored at (0,0). A) Find in terms of i and j the original position vector of the liner from the fishing boat • Exercise 17D: ‘Page 407’ Q 3

  19. Modeling A Problem - 2 [ ] -3 4 F (0,0) O L is the direction of the velocity vector Velocity is 10 km/h : v = 2 = (twice as long) [ ] [ ] -6 8 -6 8 [ ] [ ] In t hours; moved by tx -3 4 -3 4 An Ocean Liner is at (6, -6) is cruising at 10 km/h in the direction . A fishing boat is anchored at (0,0). B) Find the position vector of the liner at time t Magnitude = ((-3)2 + 42) = 25 = 5 OL = 6i - 6j+ t(-6i + 8j)

  20. Modeling A Problem - 3 [ ] -3 4 F (0,0) O L OL = 6i - 6j+ t(-6i + 8j) OL = 6i - 6j+ t(-6i + 8j) An Ocean Liner is at (6, -6) is cruising at 10 km/h in the direction . A fishing boat is anchored at (0,0). C) Find the time t when the liner is due East of the fishing boat Due East when j component is zero (-6 + 8t)j = 0 -6 + 8t = 0 8t = 6 t = 6/8 At time 3/4 hour

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