At 6 a.m. a boat A A\,A has position vector (12i−11j)(12\mathbf{i} - 11\mathbf{j})(12i−11j) km relative to a fixed origin O O\,O and moves with constant velocity (9i−6j)(9\mathbf{i} - 6\mathbf{j})(9i−6j) km h−1\text{h}^{-1}h−1. Another boat B B\,B has position vector (40i−39j)(40\mathbf{i} - 39\mathbf{j})(40i−39j) km relative to a fixed origin O O\,O and moves with constant velocity (−12i+15j)(-12\mathbf{i} + 15\mathbf{j})(−12i+15j) km h−1\text{h}^{-1}h−1.
Find expressions for the position vectors of A A\,A and BBB, in terms of t t\,t hours after 6 a.m.
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 (−18i+21j)(-18\mathbf{i} + 21\mathbf{j})(−18i+21j) km h−1\text{h}^{-1}h−1. Find the distance between the two ships at the time when they would have collided.
158 exam-style questions on OCR (MEI) A Level Maths 3.2 Kinematics, covering 3.2.1 Language of kinematics, 3.2.2 Position, displacement, distance and distance travelled, 3.2.3 Velocity, speed and acceleration distinctions, 3.2.4 Draw and interpret kinematics graphs, 3.2.5 Differentiate position and velocity (A-level only), 3.2.6 Integrate acceleration and velocity (A-level only), 3.2.7 When constant acceleration formulae apply, 3.2.8 Solve 1-D kinematics problems, 3.2.9 Language of kinematics in 2 dimensions (A-level only), 3.2.10 Extend 1-D techniques to 2-D using vectors (A-level only), 3.2.11 Cartesian equation of a path (A-level only), and 3.2.12 Vectors to solve kinematics problems (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.