Using a piecewise function we can model a force that is turned on at some moment $t=4$ and 'ramps up$\text{'}$ to a maximum, constant force of $12$ at time $t=8$ :

This piecewise function is $f(t)=\{\begin{array}{ccc}0& if& t<mn>4</mn>\\ 3(t-4)& if& 4t<mn>8</mn>\\ 12& if& t>8\end{array}$

Re$-$write this piecewise function using the unit step function $u(t-a)$ :

$y(t)=$

Then find the Laplace transform:

$Y(s)=$