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Gravitational potential energy is a fundamental concept in physics, closely related to the force of gravity. Essentially, it's the energy that an object possesses because of its position relative to a gravitational source, typically Earth.

For instance, when you hold a book above the ground, the book has gravitational potential energy due to Earth's gravity pulling it downward. The potential energy increases with height because the book has the potential to do more work if it falls from a greater distance. The mathematical formula to calculate this energy is:
\[ U = mgh \]
where \(U\) stands for gravitational potential energy, \(m\) represents the mass of the object, \(g\) is acceleration due to gravity (approximately \(9.8 m/s^2\) on Earth), and \(h\) is the height above a chosen reference point.

Interestingly, while we often consider this energy to be positive, it can be negative depending on our choice of the reference level. If our reference point is set to be the ground, any position below that would give a negative height \(h\), thus a negative gravitational potential energy. However, this concept of negative energy doesn't mean the energy is lost, it's simply a result of our frame of reference.

Imagine stretching a rubber band between your fingers. As you pull it away from its relaxed position, you're storing energy in the band. This stored energy is known as elastic potential energy.

Objects like rubber bands, springs, and even bungee cords have what we call elasticity. They can be deformed, and they return to their original shape when the force is removed due to their 'elastic properties'. The more you stretch, the more energy you're storing, up to a point. This energy can be calculated with the formula:
\[ U = \frac{1}{2}kx^2 \]
where \(U\) stands for elastic potential energy, \(k\) is a constant characteristic to the material's stiffness, known as the spring constant, and \(x\) represents the displacement from its original, undisturbed position 'the rest position'.

The key thing to note here is that the displacement is squared in this equation. This squaring guarantees that regardless of the direction of displacement (compression or extension), the elastic potential energy is always a non-negative value, ensuring you can never have negative elastic potential energy.

Kinetic energy is the energy that an object possesses due to its motion. Any moving object, from a rolling ball to a flying airplane, has kinetic energy. Faster objects or those with more mass have more kinetic energy. Unlike potential energy, which depends on an object's position or condition, kinetic energy is purely a function of motion.

The formula to calculate the kinetic energy (\(K\)) of an object is:
\[ K = \frac{1}{2}mv^2 \]
where \(m\) represents the mass of the object and \(v\) is its velocity. Since velocity is squared in the formula, kinetic energy is always a non-negative value. This is because the square of a number is always positive, ensuring that kinetic energy cannot be negative. Even if an object is moving in what we would consider a 'negative direction', the squaring of its velocity in the kinetic energy calculation negates this directional component, resulting only in a positive value.