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Question #42283

Difficulty: 8/10 | 🟢 Free

The Question

Two point particles of rest masses $m_1$ and $m_2$ are initially held at rest at a distance $R$ from each other in an inertial frame in deep space. They are released and fall towards each other under mutual gravitational attraction. Upon collision, they completely fuse into a single composite particle of rest mass $M$ , and a single photon is emitted in the process. Assuming no other energy losses and that the system is isolated, what is the energy $E$ of the emitted photon? (Here $c$ is the speed of light in vacuum, and $G$ is the gravitational constant).

Options

A.
$\frac{\left( m_1 + m_2 - \frac{G m_1 m_2}{R c^2} \right)^2 - M^2}{2 \left( m_1 + m_2 - \frac{G m_1 m_2}{R c^2} \right)} c^2$
B.
$\left( m_1 + m_2 - M - \frac{G m_1 m_2}{R c^2} \right) c^2$
C.
$\frac{(m_1 + m_2)^2 - M^2}{2 (m_1 + m_2)} c^2 - \frac{G m_1 m_2}{R}$
D.
$\frac{(m_1 + m_2)^2 - M^2}{2 M} c^2 - \frac{G m_1 m_2}{R}$

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

Gravitational Potential Energy (System)|Recoil of Atom|Mass Defect vs. Binding Energy

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of macroscopic gravitational potential energy with microscopic relativistic kinematics (mass defect and recoil).

The primary trap lies in assuming the fused mass remains at rest, leading students to simply subtract the final rest mass energy from the initial total energy ( $E = E_i - Mc^2$ ), which corresponds to Option B.

However, the emission of a single photon requires the composite mass to recoil to conserve the initially zero momentum.

By failing to account for the kinetic energy of this recoil, students select the distractor.

Furthermore, Option C traps students who incorrectly apply the gravitational potential energy as an afterthought rather than integrating it into the initial relativistic invariant mass of the system, testing their fundamental understanding of mass-energy equivalence.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Gravitational Potential Energy (System)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free