A boat must cross a river 500m wide that flows due east with uniform surface velocity 2ms−1. The boat can be rowed at 4ms−1 relative to the water. The boat is aimed so that its actual track makes a 30∘ angle north of west.
(i) Write the river-flow velocity vector and the boat’s velocity relative to the water in component form, taking east as the positive x-axis and north as the positive y-axis.
(ii) Determine the boat’s actual velocity vector with respect to the ground.
(iii) Calculate the angle between the boat’s actual velocity vector and the river-flow velocity vector, giving your answer to the nearest degree.
(iv) Using the result of part (iii), state whether the boat’s actual path is closer to being upstream or cross-stream and justify your answer quantitatively.