How to Model The Following Question as an Adiabatic Process?

This is a quite neat question, but the condition $$Vll frac{d L}{L} v_0$$ was kind of confusing to me. In retrospect I understand it, but I think this is just badly written and that makes the question hard. How I really solved the question was by thinking about the full problem and then asking how it could be simplified if we assumed that the particle was somehow moving much faster than the piston and then seeing where that takes me.

So let's get to it. The point of the adiabatic treatment is to imagine that before the piston moves even a tiny bit, the particle goes back and forth so many times one can essentially average over these collisions to describe the long-term evolution of the system. This is true if the time between the collisions is much shorter than the time it takes for the piston to hit the end of the tube. The time between collisions is $$Delta t_{col} sim frac{2L}{v}$$ where I put the approximate $sim$ since $L$ is evolving and the precise computation of $t_{col}$ would be more involved. The time for the piston to come to the end of the tube is $$T_{pist} = frac{L}{V}$$ The condition $t_{col}ll T_{pist}$ then reduces to the requirement that $V ll v$ at all times. We will see that $v$ grows in time, so it would actually be sufficient to require only $V ll v_0$ without any enigmatic reference to quantities such as "$dL/L$".

Now let us take a look at the evolution of the system. At every collision of the particle with the piston, the particle velocity grows by $2V$. You can see this by transforming into the rest-frame of the piston and treating the collision only as the particle bouncing off a static wall and then transforming back. Using the approximate formula for the time between collisions we obtain $$frac{Delta v_{coll}}{Delta t_{coll}} approx frac{V}{L} v$$ Note that this formula is valid up to other terms of relative size $V/v ll 1$. We can also relate this to the change of $L$, since during $Delta t_{coll}$ the piston will move by $Delta L_{coll} = -V Delta t_{coll}$. We then get $$frac{Delta v_{coll}}{Delta L_{coll}} approx -frac{v}{L}$$ Given the assumptions above, one can then describe the approximate evolution of the velocity smoothed over many collisions by the differential equation $$frac{d v}{dL} = -frac{v}{L}$$ This can be integrated to yield $$frac{v}{v_0} = frac{L_0}{L}$$ I believe that this sketch of the derivation gives you a good idea of how to work out the details and answer the question.