A particle moves in the \(xy\)-plane such that its position as a function of time \(t\) is given by \(\vec{r}(t) = v_0 t\,\hat{i} + \dfrac{1}{2} b t^2\,\hat{j}\), where \(v_0\) and \(b\) are positive constants. Which of the following expressions represents the magnitude of the component of the particle's acceleration perpendicular to its velocity, \(a_\perp(t)\), as a function of time \(t\)?