An autonomous trolley is modelled as a particle moving on a horizontal plane. At time t t\,t seconds, where t≥0t \ge 0t≥0, its velocity is v=2i+(2t+1)j\mathbf{v} = 2\mathbf{i} + (2t + 1)\mathbf{j}v=2i+(2t+1)j m s−1^{-1}−1.
At time t=0t = 0t=0, its position vector relative to a fixed origin O O\,O is (i+3j)(\mathbf{i} + 3\mathbf{j})(i+3j) m.
Find the position vector of the trolley at time ttt.
Find the Cartesian equation of the path of the trolley.
Give one reason why this model may become unrealistic for large values of ttt.