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Question #43519

Difficulty: 8/10 | 🟢 Free

The Question

A satellite of mass $m$ is in a circular orbit of radius $r$ around the Earth (mass $M$ ). The thin atmosphere exerts a tangential drag force of magnitude $F_d = \alpha v^2$ on the satellite, where $v$ is its orbital speed and $\alpha$ is a positive constant.

In Scenario A, the satellite's onboard engine provides a continuous tangential thrust to maintain the satellite in the exact same circular orbit of radius $r$ . Let $P_A$ be the mechanical power delivered by the engine.

In Scenario B, the engine is turned off, and the satellite's orbit slowly decays while remaining approximately circular. Let $P_B$ be the rate at which the gravitational potential energy of the satellite-Earth system decreases at the instant the orbital radius is $r$ .

What is the ratio $\frac{P_A}{P_B}$ ?

Options

A.
$\frac{1}{2}$
B.
$1$
C.
$2$
D.
$\frac{1}{4}$

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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

Geostationary Satellite Constraints|Power-Velocity Relationship|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in orbital mechanics, specifically targeting the 'satellite paradox.' A common misconception is that the rate of loss of potential energy directly equals the power dissipated by drag ( $P_B = P_{\text{drag}}$ ), which would incorrectly yield a ratio of 1.

However, because the orbit remains quasi-circular, the kinetic energy actually increases as the orbit decays ( $dK/dt > 0$ ).

Since total energy $E$ is half the potential energy $U$ , the potential energy must decrease at twice the rate of the total energy loss to account for both the work done against drag and the simultaneous increase in kinetic energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Geostationary Satellite Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free