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Question #43661

Difficulty: 8/10 | 🟢 Free

The Question

Two isolated particles of masses $m$ and $2m$ are initially at a distance $R$ from each other in free space. In the laboratory frame, they are simultaneously projected directly away from each other along the line joining them, both with a speed $v_0 = \sqrt{\frac{3Gm}{4R}}$ , where $G$ is the universal gravitational constant. What is the maximum separation $r_{\text{max}}$ between the two masses during their subsequent motion?

Options

A.
$2R$
B.
$\frac{16}{7}R$
C.
$4R$
D.
$\frac{8}{7}R$

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

Reduced Mass Effect|Center of Mass (COM) Frame|Gravitational Potential Energy (System)|Calculus-based Energy Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply energy conservation in a two-body system where the center of mass is moving.

A common trap is to apply the work-energy theorem directly in the laboratory frame, incorrectly assuming both masses momentarily come to rest at maximum separation ( $K_{\text{final}} = 0$ ).

If a student does this, they will calculate $E_{\text{lab}} = -\frac{7Gm^2}{8R}$ and arrive at the trap answer $\frac{16}{7}R$ .

However, due to momentum conservation, the system retains kinetic energy ( $K_{\text{cm}}$ ) at maximum separation.

By shifting to the Center of Mass frame and utilizing the reduced mass ( $\mu$ ), the problem simplifies to a single-particle equivalent where $K_{\text{rel}}$ correctly dictates the maximum separation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Reduced Mass Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free