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The Conservation of Energy is a fundamental principle in physics and describes how energy is neither created nor destroyed in an isolated system. For a moving particle influenced by a potential field, like the exercise example, this means that the total energy remains constant.
As the particle starts moving from coordinates (3,3) to (1,1), the initial energy—composed solely of potential energy since it starts from rest—must equal the final total energy. In this scenario, energy moves between potential and kinetic forms without any loss. This balance is described mathematically as:

• Total Initial Energy = Total Final Energy
• That implies, Potential Energy (Initial) + Kinetic Energy (Initial) = Potential Energy (Final) + Kinetic Energy (Final)

Understanding this enables us to predict that any decrease in potential energy translates into an increase in kinetic energy, keeping the energy sum unchanged. This transformation is what helps us calculate the kinetic energy at the particle's final position.

Kinetic energy is the energy of motion. When potential energy decreases due to movement in a potential field, that energy isn't lost. It is converted into kinetic energy.
According to the conservation of energy, the decrease in potential energy is exactly equal to the increase in kinetic energy.
For the given exercise, the kinetic energy at the final point is found by calculating the potential energy changes.

• Initial kinetic energy was zero since the particle was at rest initially.
• When the particle moves to coordinates (1,1), the lost potential energy becomes kinetic energy.
• The calculated change in energy is 16\(\alpha\), and therefore, the final kinetic energy equals 16\(\alpha\).

This calculation shows how kinetic energy depends on the initial and final potential energy, manifesting as the particle's motion.

Potential energy stored in a system depends on its position within a force field, like the gravitational electric fields. In this exercise, potential energy is given by the expression \(U = \alpha(x^2 + y^2)\). This indicates how energy varies with the particle's coordinates.
Here, \(\alpha\) is a constant that influences the strength of the field.
The formula means:

• The particle's potential energy increases with an increase in distance from the field's origin (0,0).
• Higher values of \(x\) and \(y\) result in greater potential energy.
• Potential energy at any point is controlled by the sum of the squares of the coordinates, weighted by \(\alpha\).

This understanding helps determine how much energy is available to be converted into other forms, such as kinetic energy when the particle moves.

Energy transformation is at the heart of this problem, where potential energy dynamically transforms into kinetic energy. This process exemplifies how mechanical energy is conserved while shifting between kinetic and potential forms.
Initially, when the particle starts at (3,3), all energy is potential. As it travels to (1,1), potential energy reduces, converted entirely to kinetic energy.

• Initial shift: energy transformation begins as the potential energy decreases.
• The potential energy drop from 18\(\alpha\) to 2\(\alpha\) illustrates a transformation path.
• The gained kinetic energy is exactly 16\(\alpha\), equating the potential energy loss.

This seamless transformation underlines the governing laws of physics, ensuring the conservation of total mechanical energy even as it moves between different forms.