Laplace II: Steps, Impulses & Convolution · Challenge problem

A spring obeys $y''+9y=A\,\delta(t-T)$ with $y(0)=1$, $y'(0)=3$. For the smallest $T>0$ at which a single hammer blow of strength $A$ can bring the mass to rest permanently ($y\equiv0$ for $t>T$), find the product $A\,T$.