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)jAt 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=10e0.5t d = 10e^{0.5t} d=10e0.5tFully 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.