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Gravitational force is the attractive pull that all objects with mass exert on each other. This fundamental force governs everything from how you stay on the ground to how planets orbit stars. In our exercise, the gravitational force acts between the Earth and the satellite. It provides the necessary centripetal force to keep the satellite in its orbit.
The formula for gravitational force between two masses, say Earth (\( M \)) and a satellite (\( m \)), is given by:

• \( F = \frac{GMm}{r^2} \)

Here, \( G \) is the gravitational constant, and \( r \) is the distance between the center of Earth and the satellite. This equation tells us that gravitational force strengthens as objects get closer. It plays a crucial role in determining both the orbital path and velocity of any object in space.
Understanding gravitational force is key to predicting and analyzing satellite motion, as well as planning satellite launches and orbits.

Orbital velocity is the speed at which a satellite must travel to maintain its orbit around a celestial body. For a satellite orbiting Earth, balancing gravitational and centrifugal forces dictates its orbital velocity.
Calculation of orbital velocity is based on the following formula:

• \( v_o = \sqrt{\frac{GM}{r}} \)

Here, \( G \) is the gravitational constant, \( M \) is Earth's mass, and \( r \) is the radius of the orbit. This formula shows that the greater the orbital radius, the lower the orbital velocity needed. This concept is crucial for understanding how satellites stay in orbit.
In our exercise, understanding the orbital velocity helps in determining how the additional velocity imparted to the particle affects its subsequent motion and minimum distance from Earth. Calculating orbital velocity serves as a foundation for understanding other dynamics of satellite motion.

Energy conservation is a critical concept when analyzing satellite motion or any mechanics problem. In a closed system like space, where the only significant force is gravity, the law of conservation of energy states that the total mechanical energy remains constant.
Mechanical energy is the sum of kinetic energy and potential energy. For our satellite problem:

• Kinetic Energy, \( KE = \frac{1}{2}mv_p^2 \)
• Potential Energy, \( PE = -\frac{GMm}{r} \)

In our example, the particle's total energy can be calculated as \( E = \frac{7GMm}{18r} \). This combines the kinetic motion's energy and the gravitational pull's energy. By calculating energies at initial and critical points, we can predict the particle's behavior in the orbit. This principle helps find variables like velocity changes and minimum distances.

To find the minimum distance from the particle to the center of the Earth after projection, we apply the principles of energy conservation.
The minimum distance, or periapsis, occurs where all components of the object's velocity become radial. At this point, all the energy is potential energy, given by:

• \( E = -\frac{GMm}{2r_{min}} \)

By equating this energy with the previously calculated total energy \( \frac{7GMm}{18r} \), we can find that the particle's new path brings it to a minimum distance \( r_{min} = \frac{r}{2} \).
This method of calculating minimum distance is important because it helps predict orbits' closest approaches, aiding in the planning of satellite trajectories and ensuring safe and effective satellite operation.