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Question #42269

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is projected from infinity with an initial speed $v_0$ and impact parameter $b$ towards a fixed force center. The potential energy of the interaction is given by $V(r) = -\frac{k}{r^3}$ , where $k > 0$ is a constant and $r$ is the radial distance from the center. To avoid spiraling into the origin, the impact parameter $b$ must be greater than or equal to a critical value $b_c$ . Find $b_c$ .

Options

A.
$\left( \frac{27 k^2}{m^2 v_0^4} \right)^{1/6}$
B.
$\left( \frac{54 k^2}{m^2 v_0^4} \right)^{1/6}$
C.
$\left( \frac{27 k^2}{2 m^2 v_0^4} \right)^{1/6}$
D.
$\left( \frac{9 k^2}{m^2 v_0^4} \right)^{1/6}$

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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

Potential vs Field Relationship|Angular Momentum conservation|Turning Point Identification

Why This Question Is Tricky

Common Mistakes

This question tests the 'Effective Potential' method for non-Keplerian central forces.

A common trap is assuming that any particle with non-zero angular momentum will naturally bypass the origin, as is the case in $1/r$ potentials where the $1/r^2$ centrifugal barrier always dominates at small distances.

However, for $V(r) = -k/r^n$ with $n > 2$ , the attractive potential overpowers the centrifugal term as $r \to 0$ , creating a finite potential barrier rather than an infinite one.

Students must correctly construct $V_{eff}$ , find its local maximum, and apply energy conservation to find the critical capture condition, forcing them to decouple from standard conic-section orbital intuition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Potential vs Field Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free