# Model of a 1-dimensional gas

**Classical Mechanics**Level 5

A block with mass \(M\) and speed \(V_{0}=1~\mbox{m/s}\), slides on a frictionless surface towards a fixed wall. The block strikes a small particle with mass \(m\), initially at rest at a distance \(L_{0}=1~\mbox{m}\) from the wall. The particle bounces elastically off the block then slides to the wall where it bounces elastically and then slides back to the block. The particle continues to bounce elastically back and forth between the block and the wall. It turns out that the block won't reach the wall. This is because the small particle can be treated as a 1-dimensional gas being compressed adiabatically. Thus, as the volume of this "gas" decreases, its pressure increases and the block eventually comes to a stop. If \(\frac{M}{m}=3\times 10^{4}\), how close **in millimeters** does the block come to the wall?