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 figure. When the distance of the piston from 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 infinitly 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 $v / L$
$(2)$ After each collision with the piston, the particle speed increases by $2 V$
$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L _0$ to $\frac{1}{2} L _0$
$(4)$ If the piston moves inward by $d L$, the particle speed increases by $2 v \frac{d L}{L}$
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 figure. When the distance of the piston from 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 infinitly 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 $v / L$
$(2)$ After each collision with the piston, the particle speed increases by $2 V$
$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L _0$ to $\frac{1}{2} L _0$
$(4)$ If the piston moves inward by $d L$, the particle speed increases by $2 v \frac{d L}{L}$
- [IIT 2019]
- A
$2,3$
- B
$2,4$
- C
$1,3$
- D
$2,4$
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- [JEE MAIN 2020]
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