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.