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WARNING: Long and perhaps boring.
Consider a long narrow cylindrical container of mass M filled with
a monatomic ideal gas with a total mass equal to that of the
cylinder. The cylindrical container has an inner length L. The
gas particles are initially distributed uniformly throughout the
volume of the cylinder. Both the container and the particles are
initially at rest in a region devoid of any "gravitational field."
You begin pushing with a constant force F on one end of the
cylinder along a direction parallel to its axis of symmetry. All
ensuing collisions between the particles and the cylinder are
elastic. Some time later it is found that the speed of the
container is v. To recap, the "givens" are M, L, F, and v.
At time t, the center of mass velocity of
the gas is the same as that of the cylinder.
At time t, the center of mass of the gas
lies at a distance L/4 from the end on which you are pushing.
i) Now can you determine the distance that the container has
moved? If so, what is it?
Yes. The container has moved a distance Mv^2/F + L/8.
(IMO, the proof of this is not at all trivial without a clear
understanding of a variety of work-energy relationships.)