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Question #42161

Difficulty: 8/10 | 🟢 Free

The Question

A solid spherical projectile of radius $R$ and emissivity $e$ moves at a constant high velocity $v$ through an atmosphere of constant density $\rho$ . The aerodynamic drag force acting on the projectile is given by $F_d = \frac{1}{2} \rho v^2 C_d A_c$ , where $C_d$ is the drag coefficient and $A_c$ is the effective cross-sectional area. Assume that the entire mechanical power dissipated by the drag force is absorbed by the projectile as internal thermal energy. If the projectile loses heat solely via thermal radiation to the surrounding environment (assumed to be at $0\text{ K}$ ), which of the following expressions represents the steady-state terminal temperature $T$ of the projectile? (Here, $\sigma$ is the Stefan-Boltzmann constant).

Options

A.
$\left( \frac{\rho v^3 C_d}{8 e \sigma} \right)^{1/4}$
B.
$\left( \frac{\rho v^3 C_d}{2 e \sigma} \right)^{1/4}$
C.
$\left( \frac{\rho v^3 C_d}{4 e \sigma} \right)^{1/4}$
D.
$\left( \frac{\rho v^2 C_d}{8 e \sigma} \right)^{1/4}$

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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

Stefan-Boltzmann Power Law|Power-Velocity Relationship|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple continuum mechanics (aerodynamic drag) with thermodynamics (Stefan-Boltzmann radiation) at steady-state equilibrium.

The primary trap lies in the geometric distinction between the areas used in the two phenomena: the drag force depends on the projected cross-sectional area ( $A_c = \pi R^2$ ), whereas thermal radiation occurs over the entire surface area of the sphere ( $A_s = 4\pi R^2$ ).

A common mistake is using the same area for both, which incorrectly cancels out the geometric factor of 4, leading to an erroneous factor of 2 in the denominator.

Additionally, students might equate force directly to radiated power instead of using mechanical power ( $P = F \cdot v$ ), leading to dimensional inconsistencies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Stefan-Boltzmann Power Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free