$A$ small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in the figure. When the distance of the piston from the closed end is $L = L_0$, the particle speed is $v = v_0$. The piston is moved inward at a very low speed $V$ such that $V \ll \frac{dL}{L} v_0$, where $dL$ is the infinitely small displacement of the piston. Which of the following statement(s) is/are correct?
$(1)$ The rate at which the particle strikes the piston is $v / (2L)$
$(2)$ After each collision with the piston, the particle speed increases by $2V$
$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L_0$ to $L_0 / 2$
$(4)$ If the piston moves inward by $dL$, the particle speed increases by $v \frac{dL}{L}$
$(1)$ The rate at which the particle strikes the piston is $v / (2L)$
$(2)$ After each collision with the piston, the particle speed increases by $2V$
$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L_0$ to $L_0 / 2$
$(4)$ If the piston moves inward by $dL$, the particle speed increases by $v \frac{dL}{L}$
- A$2, 3$
- B$2, 4$
- C$1, 3$
- D$1, 2, 3$
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$C \rightarrow A :$ Isochoric change leading to a change of pressure from $P_{2}$ to $P_{1}$.
Total work done in the complete cycle $ABCA$ is
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