A large box $(10$ kg $)$ ius placed on a flat, frictionless floor. A stick extends from the top of the box and attaches to a massless horizontal spring $(k=20\frac{N}{m}).$ The other side of the spring is attached to a smaller box $(1kg),$ which is free to slide on top of the first box. The boxes are held stationary as the spring is stretched $0\mathrm{.}50$ m $.$ Then, everything is released.

A$)$ Write the Lagrangian for the system.

B$)$ Solve $($as much as possible$)$ for the positions of both boxes.

Hint: Use $x$ for the position of box $2$ and $y$ for the position of box $1.$ Then, the spring's displacement from equilibrium will depend on the differences between these positions.