A monoatomic gas fills a cylinder of height H under a piston. The piston oscillates periodically up and down with amplitude a: during the first half-period, it moves upward with a constant speed u, and during the second half-period, it moves downward with the same constant speed u. Initially, the root-mean-square (RMS) speed of the molecules is v. How much time t will it take for the RMS speed to double? Use the following model assumptions: the walls and the piston are perfect heat insulators and have zero heat capacity; the surface of the piston is perfectly flat; the mean free path of the molecules \lambda satisfies the following conditions: H\gg\lambda\gg a and \lambda\gg \frac{H^2}{vt}; additionally, v\gg u.
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