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Question #43681

Difficulty: 8/10 | 🟢 Free

The Question

A heavy nucleus of mass $M$ initially at rest in the laboratory frame undergoes an isotropic shattering process. It breaks into a continuous distribution of fragments, converting a total mass defect $\Delta m$ into the kinetic energy of the fragments. The mass of the fragments $dm$ having a velocity magnitude between $v$ and $v+dv$ is given by the distribution $dm = k v^2 (v_0 - v) dv$ for $0 \le v \le v_0$ , where $v_0$ is the maximum velocity of any fragment and $k$ is a positive constant. Assuming $\Delta m \ll M$ and that all fragment velocities are non-relativistic, what is the maximum velocity $v_0$ of the fragments? (Here $c$ is the speed of light in vacuum).

Options

A.
$\sqrt{\frac{5 \Delta m c^2}{M}}$
B.
$\sqrt{\frac{5 \Delta m c^2}{2 M}}$
C.
$\sqrt{\frac{2 \Delta m c^2}{M}}$
D.
$\sqrt{\frac{10 \Delta m c^2}{M}}$

Check Your Answer

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

Mass Defect vs. Binding Energy|Center of Mass (COM) Frame|Integration

Why This Question Is Tricky

Common Mistakes

The core trap lies in the transition from discrete particle mechanics to continuous statistical distributions.

Students often misapply the conservation of energy by assuming the kinetic energy can be modeled as a single effective body $Q = \frac{1}{2} M v_{max}^2$ (leading to Option C) or by incorrectly averaging the velocity.

Furthermore, a common error is integrating the momentum scalar $P = \int v dm$ and then using $Q = P^2 / 2M$ , which violates the non-linear relationship between kinetic energy and velocity in a distributed system.

This question forces the rigorous application of $dK = \frac{1}{2} v^2 dm$ over the specific velocity spectrum, testing true comprehension of integral calculus applied to mass-energy equivalence.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Mass Defect vs. Binding Energy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free