Before You Begin

Question #42463

Difficulty: 8/10 | 🟢 Free

The Question

A rigid container of negligible mass ( $M_{\text{container}} \approx 0$ ) is initially filled with a monatomic ideal gas of total mass $M_0$ and temperature $T_0$ . The container is initially at rest in deep space. A small valve is opened, and the gas is expelled to the vacuum through a perfectly designed nozzle such that the exit pressure of the gas is zero. Assume the gas remaining in the container undergoes a reversible adiabatic process, and the flow through the nozzle is steady and isentropic. Find the velocity of the container when the mass of the gas remaining inside is $\frac{M_0}{8}$ . (Let $R$ be the universal gas constant and $\mu$ be the molar mass of the gas).

Options

A.
$\frac{3}{2} \sqrt{\frac{5 R T_0}{\mu}}$
B.
$3 \ln(2) \sqrt{\frac{5 R T_0}{\mu}}$
C.
$\frac{3}{2} \sqrt{\frac{3 R T_0}{\mu}}$
D.
$\frac{9}{8} \sqrt{\frac{5 R T_0}{\mu}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Recoil and Thrust Forces|Adiabatic Process Constraint|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Collision' between variable-mass dynamics and thermodynamics.

Students typically memorize the Tsiolkovsky rocket equation $v = v_e \ln(M_0/M)$ and apply it blindly, assuming a constant exhaust velocity (Option B).

However, as the gas escapes, the remaining gas undergoes an adiabatic expansion, causing its temperature and internal energy to drop.

Since the exhaust kinetic energy derives from the gas enthalpy ( $v_e = \sqrt{2 c_p T}$ ), $v_e$ decreases continuously.

This requires coupling the thermodynamic state of the gas with Newton's second law, leading to a modified differential equation $dv = -v_e(M) dM/M$ that must be integrated, perfectly blending the Work-Energy Theorem with Variable Mass constraints.

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