An unsteady fluid flow has velocity field given in Cartesian coordinates $(x, y, z)$ by $\mathbf{u}=(1, x t, 0)$, where $t$ denotes time. Dye is released into the fluid from the origin continuously. Find the position at time $t$ of the dye particle that was released at time $s$ and hence show that the dye streak lies along the curve

$$y=\frac{1}{2} t x^{2}-\frac{1}{6} x^{3}$$