Compressing A Single-Particle Gas

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., vu1,u2v\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 vv and the distance between the pistons LL. Initially, the speed of the particle is v0=300 m/sv_{0}=300~\text{m}/\text{s} and the distance between the pistons is L0=10 cmL_{0}=10~\text{cm}. Find the speed of the particle in meters per second when the distance between the pistons becomes L03.\frac{L_{0}}{3}.

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