Two drones, Phoenix and Icarus, are performing a synchronized survey of a forest fire. Both drones fly in the same direction along straight, parallel paths at a constant altitude.
Phoenix maintains a constant velocity of (3i+4j) m s−1(3\mathbf{i} + 4\mathbf{j})\text{ m s}^{-1}(3i+4j) m s−1.
At time t=0t = 0t=0 seconds, Icarus is at the position (7i−24j) metres(7\mathbf{i} - 24\mathbf{j})\text{ metres}(7i−24j) metres and is moving with a constant speed of 15 m s−115\text{ m s}^{-1}15 m s−1.
(i) Explain why Icarus's velocity must be of the form k(3i+4j) m s−1k(3\mathbf{i} + 4\mathbf{j})\text{ m s}^{-1}k(3i+4j) m s−1, where kkk is a constant.
(ii) Show that k=3k = 3k=3.
Determine the position vector of Icarus when t=2t = 2t=2.
At both t=0t = 0t=0 and t=2t = 2t=2, the distance between Phoenix and Icarus is 26 metres26\text{ metres}26 metres.
Calculate the shortest distance between their two parallel lines of flight.
Fully justify your answer.
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.