Consider the initial value problem

$$\mathcal{L} x(t)=f(t), \quad x(0)=0, \quad \dot{x}(0)=0, \quad t \geqslant 0,$$

where $\mathcal{L}$ is a second-order linear operator involving differentiation with respect to $t$. Explain briefly how to solve this by using a Green's function.

Now consider

$$\ddot{x}(t)= \begin{cases}a & 0 \leqslant t \leqslant T \\ 0 & T<t<\infty\end{cases}$$

where $a$ is a constant, subject to the same initial conditions. Solve this using the Green's function, and explain how your answer is related to a problem in Newtonian dynamics.