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.
Practise Edexcel A Level Maths Further Kinematics with exam-style questions for A Level Maths. 54 questions covering Vectors in Kinematics, Vector Methods with Projectiles, Variable Acceleration in One Dimension, Differentiating Vectors, and Integrating Vectors, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.