A train travels along a straight, horizontal track aligned with the \(+x\)-axis with a time-dependent acceleration given by \(a_x(t) = bt\), where \(b\) is a positive constant. The train is at rest at time \(t = 0\). Due to horizontal wind shear, the velocity of falling raindrops relative to the ground is \(\vec{v}_{r/g}(t) = (u_0 + ct)\hat{i} - v_0\hat{j}\), where \(u_0\), \(c\), and \(v_0\) are positive constants, and \(\hat{j}\) points vertically upward. In terms of the given quantities, at what time \(t > 0\) do the raindrops appear to fall purely vertically to an observer on the train?