Vectors (10) Modelling

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

Velocity Vector Calculation (1) s = 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 A + s = B r s = - r A B B = 2i - 4j s = (2i - 4j) - (5i + 5j) B s = 2i - 4j - 5i - 5j s = -3i - 9j

Velocity Vector Calculation (2) s = 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 average velocity = change in position vector time taken r B = 2i - 4j = (-3i - 9j) m 3 sec. average velocity B average velocity = (-i - 3j) ms-1 Meaning 1 ms-1 West and 3 ms-1 South

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

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

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

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

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

Resultant Velocity (4) v = 3i + 3j + 5i - 2j 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 resultant velocity v

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

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. O P 50km OP = 700i + 500j 70km

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 OP = 700i + 500j O

1 hour later 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 PC = 300i - 220j OP = 700i + 500j C O OC = 700i + 500j + 300i - 220j OC = 1000i + 280j 1 hour later

2 hours later 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 PC = 600i - 440j OP = 700i + 500j C O OC = 700i + 500j + 600i - 440j OC = 1300i + 60j 2 hours later

5 hours later 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 OP = 700i + 500j PC = 5(300i - 440j) O OC = 700i + 500j + 1500i - 2200j OC = 2200i - 700j C 5 hours later

t hours later 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 PC = t(300i - 220)j OP = 700i + 500j Distance = speed x time C O OC = 700i + 500j + t(300i - 220j) Original Point Parallel Vector t hours later

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

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

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