Here are step-by-step instructions. There is a small parameter, u/v\ll 1; in your calculations, you may drop any terms which have u/v\ll 1 in a power larger than two. Calculate the increase in the molecule’s energy during one collision \mathrm dK_\pm with the piston, depending on the x-component of the velocity of the molecule w and the direction of motion of the piston. Using the Maxwell distribution and the total number of molecules N (it will cancel out from the answer), express the frequency of collisions with the approaching piston \mathrm df_+ and with the departing piston \mathrm df_- for molecules for which the x-component of the velocity is within the range [w,w+\mathrm dw]. Find the energy change of molecules \Delta K=\Delta t\left(\int K_+\mathrm df_++\int K_-\mathrm df_-\right). This expression includes the temperature. Express T in terms of the root-mean-square three-dimensional speed v. Relate \Delta K to the internal energy change of the gas (expressed in terms of v). Solve this differential equation to find how v depends on time.
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