A micro-drone is being tested in a flight chamber. Its velocity v m s−1\mathbf{v} \text{ m s}^{-1}v m s−1 at time t t\,t seconds is modelled by the vector function:
v=(5e0.5tsin2t+20e0.5tcos2t)i+(5e0.5tcos2t−20e0.5tsin2t)j\mathbf{v} = (5 e^{0.5t} \sin 2t + 20 e^{0.5t} \cos 2t) \mathbf{i} + (5 e^{0.5t} \cos 2t - 20 e^{0.5t} \sin 2t) \mathbf{j}v=(5e0.5tsin2t+20e0.5tcos2t)i+(5e0.5tcos2t−20e0.5tsin2t)j
At time t=0t = 0t=0, the drone is at the point with position vector 10j10\mathbf{j}10j metres.
Determine the distance, d d\,d metres, of the drone from the origin at time t t\,t and show that it is given by:
d=10e0.5td = 10e^{0.5t}d=10e0.5t
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.