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Question #42264

Difficulty: 8/10 | 🟢 Free

The Question

A space probe of initial mass $m_0$ is released from rest at a distance $R_0$ from the center of a planet of mass $M$ . As it falls radially inwards under the planet's gravity, it sweeps up a stationary cloud of cosmic dust. The mass of the probe increases such that its mass at a distance $r$ from the center is given by $m(r) = m_0 \frac{R_0}{r}$ . Assuming the dust particles are initially at rest with respect to the planet and the accretion is completely inelastic, what is the speed of the probe when it reaches a distance $R$ ( $R < R_0$ ) from the center? (Ignore any atmospheric drag or forces other than gravity).

Options

A.
$\sqrt{ \frac{2GM(R_0^3 - R^3)}{3 R_0^3 R} }$
B.
$\sqrt{ \frac{2GM(R_0^2 - R^2)}{R_0^2 R} }$
C.
$\sqrt{ \frac{2GM(R_0 - R)}{R_0 R} }$
D.
$\sqrt{ \frac{GM(R_0^2 - R^2)}{R_0^3} }$

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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

Gravitational Potential Reference|Variable Mass Systems|Collision Impulses

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between conservative systems and variable mass systems involving inelastic accretion.

A common trap is applying the Work-Energy Theorem or conservation of mechanical energy directly ( $E_i = E_f$ ), which incorrectly assumes that the work done by gravity solely changes the kinetic and potential energy of the instantaneous mass.

However, sweeping up stationary dust constitutes a perfectly inelastic collision where kinetic energy is continuously lost to internal heat.

Students must recognize that $F_{ext} = dp/dt$ is the fundamental principle here, and correctly set up the differential equation for momentum with respect to radial distance to account for the momentum flux.

Quick Facts About This Question

Question Information

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