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Question #42462

Difficulty: 8/10 | 🟢 Free

The Question

A thermally insulated vessel of mass $M$ contains a monoatomic ideal gas of total mass $m_0$ (where $m_0 \ll M$ ) and molar mass $\mu$ at an initial temperature $T_0$ . The vessel is initially at rest in gravity-free space. A small hole of area $A$ is opened, allowing the gas to effuse out. The thrust force experienced by the vessel due to effusion is $F = \frac{1}{2} P A$ , where $P$ is the instantaneous pressure inside the vessel. Assuming each effusing molecule carries away an average kinetic energy of $2kT$ , what is the final speed of the vessel after all the gas has escaped?

Options

A.
$\frac{6}{7} \frac{m_0}{M} \sqrt{\frac{\pi R T_0}{2 \mu}}$
B.
$\frac{m_0}{M} \sqrt{\frac{\pi R T_0}{2 \mu}}$
C.
$\frac{3}{4} \frac{m_0}{M} \sqrt{\frac{\pi R T_0}{2 \mu}}$
D.
$\frac{2}{3} \frac{m_0}{M} \sqrt{\frac{\pi R T_0}{2 \mu}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Recoil and Thrust Forces|Effusion and Diffusion|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of variable mass dynamics and kinetic theory.

A common lethal trap is assuming the gas undergoes standard reversible adiabatic expansion ( $T \propto N^{\gamma-1}$ ), leading to an incorrect integration factor of 3/4.

Another trap is ignoring the effusion cooling effect entirely (assuming isothermal conditions), leading to a factor of 1.

The correct approach requires applying energy conservation to the effusing molecules ( $dU = 2kT dN$ ) to find the true temperature dependence $T \propto N^{1/3}$ , and then integrating the time-dependent exhaust velocity $v_{ex} \propto T^{1/2} \propto m^{1/6}$ in the rocket equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Recoil and Thrust Forces
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free