JEE Advanced · Physics · 6. Work Power Energy

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 $V < < \frac{d L}{L} v_{0},$ where $d L$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

A The rate at which the particle strikes the piston is $\frac{v}{L}$
B After each collision with the piston, the particle speed increases by $2 V$
C The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L_{0}$ to $\frac{1}{2} L_{0}$
D If the piston moves inward by $d L,$ the particle speed increases by $2 v \frac{d L}{L}$

Verified Solution

Answer & Solution

Correct Answer

(C) The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L_{0}$ to $\frac{1}{2} L_{0}$

Step-by-step Solution

Detailed explanation

Therefore change in speed

= (2 V + V_{0} - V_{0})

= 2 V

In every collision it acquires

2 V \Rightarrow (B)

is correct

Now, frequency of collision of particle with piston,

\Rightarrow f = \frac{V}{2 x}

(as it has to travel

“ 2 x ”

distance with speed

“ V ”

)

\Rightarrow (A)

is incorrect
Acceleration

\Rightarrow \frac{d v}{d t} = f \times 2 V

\Rightarrow d v = \frac{V}{2 x} 2 V d t

d v = \frac{V}{2 x} 2 (- d x) (a s V d t = - d x)

\int_{V_{0}}^{V} \frac{d v}{v} = \int_{l}^{x} \frac{- d x}{x}

\Rightarrow l n \frac{V}{V_{0}} = - l n \frac{x}{l}

\Rightarrow V = \frac{V_{0} l}{x}

where

x = \frac{l}{2}, V = 2 V_{0}

∴ K_{i} = \frac{1}{2} m V_{0}^{2}

and

K_{f} = \frac{1}{2} m {(2 V_{0})}^{2} = 4 . K_{i}

\Rightarrow \frac{K_{f}}{K_{i}} = 4 \Rightarrow (C)

is correct.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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