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Question #85154

Difficulty: 8/10 | 🟢 Free

The Question

A spacecraft contains a highly rarefied ideal gas of molar mass $M_0$ , uniform mass density $\rho$ , and temperature $T$ . Inside, a thin, perfectly reflecting free-floating partition of mass $M$ and area $A$ is oriented perpendicular to the spacecraft's longitudinal axis. Immediately after the spacecraft begins to accelerate at a constant rate $a$ along its axis, the partition reaches a quasi-steady terminal velocity $u$ ( $u \ll v_{rms}$ ) relative to the spacecraft. Assume the gas remains uniform and its velocity distribution is Maxwellian in the spacecraft's frame. Which of the following expressions correctly represents $u$ ?

Options

A.
$\frac{M a}{4 \rho A} \sqrt{\frac{\pi M_0}{2 R T}}$
B.
$\frac{\sqrt{3} M a}{4 \rho A} \sqrt{\frac{M_0}{R T}}$
C.
$\frac{M a}{4 \rho A} \sqrt{\frac{\pi M_0}{R T}}$
D.
$\frac{M a}{2 \rho A} \sqrt{\frac{\pi M_0}{2 R T}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

RMS vs Average vs Most Probable Speed|Mean Free Path|Collision Impulses

Why This Question Is Tricky

Examiner Levers

Pattern Traps

Forces student to avoid "obvious" but wrong patterns or sequences.

Difficulty Boost : +2

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to correctly apply kinetic theory to a moving macroscopic target in a non-inertial frame.

A common trap is using the macroscopic pressure formula $P = \frac{1}{3}\rho v_{rms}^2$ and substituting $v_{rms} \pm u$ , which incorrectly assumes all molecules move at exactly $v_{rms}$ (leading to Option B).

Another trap is blindly applying the $\sqrt{2}$ relative velocity factor from the mean free path derivation (leading to Option C), failing to realize that the $\sqrt{2}$ factor only applies to collisions between identical gas molecules, not a macroscopic partition.

The correct approach requires integrating the momentum transfer over the 1D Maxwellian velocity distribution, yielding a drag force proportional to the mean speed $\langle v \rangle$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	RMS vs Average vs Most Probable Speed
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free