The additional K.E. to be provided to a satellite of mass m revolving around a planet of mass M,

step 1 Energy required to move a body from off mass M from the orbit of the radius to R to 3R. So energ

The additional K.E. to be provided to a satellite of mass m...

The additional K.E. to be provided to a satellite of mass $\mathrm{m}$ revolving around a planet of mass $\mathrm{M}$, to transfer it from a circular orbit of radius $\mathrm{R}_{1}$ to another radius $\mathrm{R}_{2}\left(\mathrm{R}_{2}>\mathrm{R}_{1}\right)$ is (A) $\operatorname{GMm}\left[\left(1 / R_{1}^{2}\right)-\left(1 / R_{2}^{2}\right)\right]$ $\operatorname{GMm}\left[\left(1 / R_{1}\right)-\left(1 / R_{2}\right)\right]$ (C) $2 \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]$ (D) $(1 / 2) \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]$

The additional K.E. to be provided to a satellite of mass $\mathrm{m}$ revolving around a planet of mass $\mathrm{M}$, to transfer it from a circular orbit of radius $\mathrm{R}_{1}$ to another radius $\mathrm{R}_{2}\left(\mathrm{R}_{2}>\mathrm{R}_{1}\right)$ is
(A) $\operatorname{GMm}\left[\left(1 / R_{1}^{2}\right)-\left(1 / R_{2}^{2}\right)\right]$
$\operatorname{GMm}\left[\left(1 / R_{1}\right)-\left(1 / R_{2}\right)\right]$
(C) $2 \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]$
(D) $(1 / 2) \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]$

Key Concepts

Energy Transfer in Orbital Transitions

When transferring a satellite from one circular orbit to another, additional energy must be provided to change the orbital parameters. This energy corresponds to the difference in total mechanical energy between the initial and final orbits. Accurately calculating this energy difference is critical in mission design and orbital maneuvers, ensuring that the satellite reaches the desired orbit with the correct velocity.

Total Mechanical Energy of Orbits

The total mechanical energy of an object in a circular orbit is the sum of its kinetic and gravitational potential energies. For circular orbits, this total energy has a specific relation, being equal to negative one-half of the gravitational potential energy (or -GMm/2R). The energy difference between two orbits, therefore, can be computed directly from the respective orbital energies.

Kinetic Energy in Circular Orbits

For an object in circular orbit, kinetic energy is determined by its orbital velocity, which is obtained by balancing gravitational force with the required centripetal force. The orbital speed depends on the central mass and the orbit radius, so the kinetic energy, given by one half the mass times the square of the velocity, decreases as the orbit radius increases.

Gravitational Potential Energy

In orbital mechanics, gravitational potential energy represents the energy due to the position of an object in a gravitational field. It is usually negative (indicating a bound state) and depends inversely on the distance from the central mass. As an object moves to a higher orbit (greater distance), its gravitational potential energy becomes less negative, meaning energy must be supplied to counteract the gravitational attraction.

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