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Question #42146

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The Question

A lead bullet of mass $m$ , specific heat capacity $c$ , surface area $A$ , and emissivity $e$ is fired into a large target. The bullet enters the target at $t=0$ with an initial velocity $v_0$ and experiences a constant retarding force $F$ . It comes to rest at time $t = t_0$ . The initial temperature of the bullet and the constant temperature of the surroundings are both $T_0$ . Assuming all the kinetic energy lost by the bullet is continuously converted into heat within the bullet, and it simultaneously radiates heat to the surroundings according to the Stefan-Boltzmann law, which of the following expressions gives the exact net temperature rise $\Delta T = T(t_0) - T_0$ of the bullet? (Assume the bullet does not melt, $T(t)$ is its temperature at time $t$ , and $\sigma$ is the Stefan-Boltzmann constant).

Options

A.
$\frac{v_0^2}{2c} - \frac{e \sigma A}{mc} \int_{0}^{t_0} (T(t)^4 - T_0^4) dt$
B.
$\frac{v_0^2}{2c} - \frac{e \sigma A}{mc} (T(t_0)^4 - T_0^4) t_0$
C.
$\frac{v_0^2}{2c}$
D.
$\frac{v_0^2}{2c} - \frac{e \sigma A}{mc} \int_{0}^{t_0} T(t)^4 dt$

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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

Calorimetry with Heat Loss|Work Done by Friction|Mechanical Energy Conservation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to formulate a non-linear energy balance equation involving both mechanical work and radiative heat transfer.

The primary trap is the 'adiabatic assumption' (Option C), where students incorrectly assume that all lost kinetic energy translates directly into a temperature rise ( $\Delta T = v_0^2 / 2c$ ) without accounting for the continuous heat loss during the penetration time $dt$ .

Another common mistake is treating the radiative loss as a constant evaluated at the final temperature (Option B) or forgetting the ambient temperature term in the Stefan-Boltzmann law (Option D).

By requiring the formal integral, the problem forces students to recognize that temperature rise is governed by the instantaneous difference between frictional heating and radiative cooling rates.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Calorimetry with Heat Loss
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free