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Question #43793

Difficulty: 8/10 | 🟢 Free

The Question

A nucleus, initially at rest in an excited state, transitions to its ground state by emitting a gamma-ray photon. The rest mass of the nucleus in its ground state is $M$ . The energy equivalent of the mass difference between the excited state and the ground state is $\Delta E$ . Taking all relativistic effects into account, what is the exact recoil velocity $v$ of the nucleus after the emission? (Here $c$ is the speed of light in vacuum)

Options

A.
$c \left[ \frac{(\Delta E)^2 + 2Mc^2\Delta E}{(\Delta E)^2 + 2Mc^2\Delta E + 2(Mc^2)^2} \right]$
B.
$c \left[ \frac{\Delta E}{Mc^2} \right]$
C.
$c \left[ \frac{(\Delta E)^2 + 2Mc^2\Delta E}{(\Delta E)^2 + 2Mc^2\Delta E + (Mc^2)^2} \right]$
D.
$c \left[ \frac{2Mc^2\Delta E}{(\Delta E)^2 + 2(Mc^2)^2} \right]$

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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

Radius and Velocity Scaling|Recoil of Atom|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question targets a common trap where students apply classical mechanics to a relativistic domain.

Students often incorrectly assume the photon energy is exactly $\Delta E$ and use the classical recoil formula $v = \Delta E / Mc$ , completely ignoring the recoil energy loss.

Even if they account for the recoil energy classically, they fail to use the exact relativistic energy-momentum invariant $E^2 = (pc)^2 + (Mc^2)^2$ and the relativistic velocity relation $v = pc^2/E$ .

By forcing the rigorous application of relativistic conservation laws, this problem differentiates students who memorize approximations from those who understand fundamental momentum-energy coupling.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Radius and Velocity Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free