Two identical satellites are at R and 7 R away from each surface, the wrong statement is ( R - R

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Two identical satellites are at R and 7 R away from each...

Two identical satellites are at $\mathrm{R}$ and $7 \mathrm{R}$ away from each surface, the wrong statement is ( $\mathrm{R}$ - Radius of earth) (A) ratio of total energy will be 4 (B) ratio of kinetic energies will be 4 (C) ratio of potential energies will be 4 (D) ratio of total energy will be 4 but ratio of potential and kinetic energies will be 2

Two identical satellites are at $\mathrm{R}$ and $7 \mathrm{R}$ away from each surface, the wrong statement is ( $\mathrm{R}$ - Radius of earth)
(A) ratio of total energy will be 4
(B) ratio of kinetic energies will be 4
(C) ratio of potential energies will be 4
(D) ratio of total energy will be 4 but ratio of potential and kinetic energies will be 2

Key Concepts

Energy Ratio Analysis in Orbital Mechanics

This key concept involves comparing the energies of satellites at different orbital radii using the derived expressions. By examining the ratios of kinetic, potential, and total energies, one can determine how energy scales with distance in a gravitational field. Such analyses help in assessing the validity of various statements regarding the energetic properties of satellites in different orbits.

Total Orbital Energy

The total energy of a satellite in a circular orbit is the sum of its kinetic and potential energies and is given by E = -GMm/(2r). This expression shows that the total energy is negative, indicating a bound system, and varies inversely with the orbital radius. It integrates both the kinetic and potential components in a simple, radius-dependent form.

Circular Orbit Dynamics

This concept revolves around the fact that a satellite in a circular orbit moves at a constant speed because the gravitational force acts as the necessary centripetal force. It leads to the derivation of orbital velocity, where v² is proportional to GM/r, illustrating the balance between gravitational pull and inertial motion.

Gravitational Potential Energy

Gravitational potential energy in a satellite system is given by U = -GMm/r, showing an inverse proportionality to the radial distance from the central body. This concept is crucial for understanding how changes in orbital radius affect the energy state of the satellite, with larger orbits corresponding to less negative potential energy values.

Kinetic Energy in Orbital Motion

In the context of circular orbits, the kinetic energy of a satellite is determined by its orbital speed, expressed as K = GMm/(2r). This relationship arises due to the equilibrium condition where the gravitational force provides the required centripetal force for circular motion, implying that kinetic energy is also inversely proportional to the orbital radius.

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