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Gravitational potential energy is like an invisible link tethering an object to a celestial body due to gravity. It's the energy an object possesses because of its position in a gravitational field. For an object near the Earth's surface, the gravitational potential energy (GPE) can be calculated using the formula:\[ U = -\frac{GMm}{R} \]- **\(G\):** The universal gravitational constant.- **\(M\):** The mass of Earth.- **\(m\):** The mass of the object.- **\(R\):** The distance from Earth's center to the object. For a surface-bound object, this is simply Earth's radius.This negative sign indicates that as the object gets further from Earth, it gains potential energy. Imagine pulling a bungee cord taut; you're storing energy in the stretched cord. Similarly, moving an object further into space (away from Earth), energy is stored in its position due to gravity. The further you go, the more energy is stored (or required to reach there). This energy becomes particularly important when discussing escape velocity, which is the energy at infinity, where gravitational effects eviscerate.

Kinetic energy is what you observe when you see an object moving. It's the energy of motion and is directly proportional to both the mass of the object and the square of its velocity. You can think of kinetic energy as the driving force of a speeding bullet or a fast-moving car. The formula for kinetic energy (\[ KE = \frac{1}{2}mv^2 \]reveals that even a small increase in velocity translates to a significantly higher kinetic energy. This relationship emphasizes the importance of velocity, which is why achieving escape velocity requires a precise amount of kinetic energy. - **\(m\):** The mass of the object.- **\(v\):** The velocity of the object.In our exercise, a body is projected from Earth's surface with a velocity of \(2v_e\), doubling the escape velocity. This means the kinetic energy at launch is quadruple the minimum required to escape Earth's gravity. Such excess kinetic energy indicates that when this body reaches infinity (beyond gravitational pull), it still retains positive velocity. This extra energy manifests as non-zero velocity at infinity.

The conservation of energy is a fundamental principle that assures us energy in a closed system stays constant over time. This means that the total energy—considering both kinetic and potential forms—remains the same unless some external force outside of the system intervenes. For our exercise, this principle is expressed mathematically as:- **Initial Energy (on Earth):** Sum of the initial kinetic and potential energies.- **Final Energy (at infinity):** The kinetic energy is non-zero, but potential energy is considered negligible as gravitational effects vanish.The principle can be set into an equation as:\[\frac{1}{2}mv_i^2 - \frac{GMm}{R} = \frac{1}{2}mv_\infty^2 + 0\]To find \(v_\infty\) or the velocity at infinity, we apply conservation of energy. Substituting known values simplifies calculations, showing that though potential energy becomes negligible at infinity, the initial energy surplus allows the object to retain speed even in space. This showcases why conservation of energy is pivotal in determining the motion and final velocity of objects influenced by gravitational forces.