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Question #42583

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The Question

A steady, parallel beam of gas particles, each of mass $m$ , is injected at $x=0$ with an initial number density $n_0$ and a uniform velocity $v_0$ in the $+x$ direction. The beam travels towards a perfectly elastic wall. At a given instant, the wall is located at $x=d$ and is moving with a constant velocity $u$ ( $u < v_0$ ) in the $+x$ direction. The mean free path of the particles in the region $0 \le x \le d$ is $\lambda$ . Assuming any particle that undergoes a collision is scattered out of the beam and does not reach the wall, what is the pressure exerted on the wall by the unscattered beam particles at this instant?

Options

A.
$2 n_0 m (v_0 - u)^2 e^{-d/\lambda}$
B.
$2 n_0 m v_0 (v_0 - u) e^{-d/\lambda}$
C.
$2 n_0 m (v_0 - u)^2$
D.
$n_0 m (v_0 - u)^2 e^{-d/\lambda}$

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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

Mean Free Path|Impulse-Momentum Theorem|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the Kinetic Theory of Gases with the Knudsen regime transition and relative motion.

The primary trap is ignoring the mean free path attenuation (Option C), which assumes the ideal KTG limit where $\lambda \to \infty$ .

A secondary trap involves incorrectly calculating the collision flux or momentum transfer by failing to use the relative velocity $v_0 - u$ for both the collision rate and the impulse (Option B).

Students must recognize that both the effective density at the boundary layer and the Galilean transformation of the collision parameters dictate the final pressure.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Mean Free Path
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free