Question 9 A container ship is observed to be sailing at a constant velocity v . Over an interval of 60 minutes the ship is observed to have moved to a point B which is b metres north and metres east of the initial observation at A, as shown in Figure 1. i : Figure 1 Ship sails from A to B {a) Calculate components of the ship's velocity, in m ! with respect to due north and due east if b = 43.2 km and = 18.0 km. (2 marks) {b) Calculate the magnitude of the ship's velocity |v,| in m s~ and the angle that the ship's trajectory makes with respect to the direction due north. {4 marks) On a separate occasion the ship makes a different journey. This time the ship's velocity, v, with respect to the surrounding land is the sum of two components: the ship's velocity through the sea v, and the velocity of the water cumrent in which the ship is sailing v, as shown in Figure 2. L& Figure 2 The ship's velocity v, is the sum of the ship's velocity v, and the velocity of the water cumrent v, (e} Write the vector eqguation that expresses v;in terms of v, and v, (1 mark) (d) Using the Cartesian unit vectors iand j, express the components of v, in terms of v,, v, & and , where v, and v, are the magnitudes of v, and vy, respectively. Write two equations, expressing vy and v, (the i and j components of v ) in terms of i, 1, 8 and . (8 marks) (&) Write the two equations in matrix form. {3 marks) (f) Calculate the inverse of the 2 x 2 matrix and use this to form an expression for the magnitudes of v; and v,,. (6 marks) {g) Given the components 1, =7 ms~* and 1, =4 ms~* and the angles & = 107 and = 60, calculate the ship's speed through the sea and the water current speed. Cuote your final results to two significant figures. (6 marks)