Sketch the graph of a cubic function that has three distinct real roots and a positive coefficient of $x^3$.

The vertical displacement, $s$, of a mechanical component is modelled by the function

where $t\ge 0$ is time, and $a$ and $k$ are constants with $a>0$.

Show that the curve $s(t)$ has a stationary point at its $s$-intercept.

Given that the polynomial equation $s(t)=0$, when extended to all real values of $t$, has three distinct real roots, determine the range of possible values for $k$ in terms of $a$. You must use the second derivative to justify the nature of the stationary points used in your calculation.