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Question #42273

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The Question

A comet of mass $m$ is in a parabolic orbit around a massive star of mass $M$ ( $M \gg m$ ). At its perihelion distance $r_0$ , it passes through a localized, stationary cloud of cosmic dust. The comet sweeps up a mass $\Delta m = \frac{m}{3}$ from the cloud in a completely inelastic collision. The energy dissipated during this process is entirely absorbed by the comet, exciting its internal vibrational modes. Which of the following gives the energy $E_{vib}$ absorbed by the vibrational modes and the new semi-major axis $a'$ of the comet's orbit?

Options

A.
$\displaystyle E_{vib} = \frac{GMm}{4r_0}, \quad a' = \frac{8r_0}{7}$
B.
$\displaystyle E_{vib} = \frac{GMm}{4r_0}, \quad a' = \frac{6r_0}{7}$
C.
$\displaystyle E_{vib} = \frac{GMm}{3r_0}, \quad a' = \frac{8r_0}{7}$
D.
$\displaystyle E_{vib} = \frac{GMm}{3r_0}, \quad a' = \frac{6r_0}{7}$

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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

Kepler’s Second Law (Areal Velocity)|Internal Energy Dependency|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of conservation laws in variable-mass orbital mechanics.

A common trap is assuming the dissipated energy is simply the kinetic energy of the accreted mass evaluated in the comet's initial frame ( $\frac{1}{2}\Delta m v_0^2 = \frac{GMm}{3r_0}$ ), which ignores the reduced mass of the system.

Furthermore, students often forget to use the updated mass $m'$ in the semi-major axis formula $E = -GMm'/2a$ , incorrectly yielding $a' = 6r_0/7$ .

Finally, it highlights that while angular momentum is conserved during the collision, the areal velocity $dA/dt = L/2m$ strictly decreases due to the mass increase, breaking standard Keplerian assumptions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Kepler’s Second Law (Areal Velocity)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free