At 6 a.m. a boat A A\,A has position vector (10i−8j)(10\mathbf{i} - 8\mathbf{j})(10i−8j) km relative to a fixed origin O O\,O and moves with constant velocity (8i−4j)(8\mathbf{i} - 4\mathbf{j})(8i−4j) km h−1\text{h}^{-1}h−1. Another boat B B\,B has position vector (31i−29j)(31\mathbf{i} - 29\mathbf{j})(31i−29j) km relative to a fixed origin O O\,O and moves with constant velocity (−6i+10j)(-6\mathbf{i} + 10\mathbf{j})(−6i+10j) km h−1\text{h}^{-1}h−1.
Show that if both boats maintain their course and speed, they will collide and find the time and position vector at which this occurs.
At 7 a.m. boat A A\,A realises that a collision is imminent and changes course so that it now has velocity (−10i+16j)(-10\mathbf{i} + 16\mathbf{j})(−10i+16j) km h−1\text{h}^{-1}h−1. Find the distance between the two ships at the time when they would have collided.
363 exam-style questions on Edexcel A Level Maths Further Kinematics, covering 8.1 Vectors in Kinematics, 8.2 Vector Methods with Projectiles, 8.3 Variable Acceleration in One Dimension, 8.4 Differentiating Vectors, and 8.5 Integrating Vectors. Each one has a worked solution and a mark scheme showing where the marks go.