Laplace II: Steps, Impulses & Convolution · Challenge problem

A body at rest is pushed by a triangular force pulse: $y''=f(t)$, $y(0)=y'(0)=0$, with $f(t)=t$ on $[0,1)$, $f(t)=2-t$ on $[1,2)$, and $f(t)=0$ for $t\ge2$. Using $\mathcal{L}\{f\}=\dfrac{(1-e^{-s})^2}{s^2}$, find $y(3)$.