# Compressing A Single-Particle Gas

**Classical Mechanics**Level 5

Consider a particle moving between two pistons as shown in the figure. The distance between the pistons is reduced slowly (as slowly as you want), i.e., \(v\gg u_{1}, u_{2}\) and the collisions are perfectly elastic. In this problem, we want to determine the equation of state relating the speed of the particle \(v\) and the distance between the pistons \(L\). Initially, the speed of the particle is \(v_{0}=300~\text{m}/\text{s}\) and the distance between the pistons is \(L_{0}=10~\text{cm}\). Find the speed of the particle **in meters per second** when the distance between the pistons becomes \(\frac{L_{0}}{3}.\)