The equation of motion of a particle is given in Cartesian coordinates by the relation \(\mathbf{r}(t)=\alpha t^{2} \hat{\mathbf{i}}+\beta t^{2} \hat{\mathbf{j}}\). Determine (i) the trajectory of the motion and (ii) the scalar function \(s(t)\) indicating the distance traveled as time varies. The initial condition at \(t=0\) is \(s(0)=0\). Verify the dimensional correctness of the equations found.