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Gravitational potential energy is a form of energy that an object possesses due to its position in a gravitational field. It is directly related to the object's mass, the gravitational force, and the height from which it is elevated.
This type of energy can be calculated using the formula \( \Delta U_g = mgh \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, typically taken as \( 9.8 \ \mathrm{m/s^2} \), and \( h \) is the height.
By understanding gravitational potential energy, students can grasp how energy conservation plays out in vertical movements. For example, as a marble fired from a spring gun ascends, its potential energy increases while converting from its initial kinetic and elastic potential energy.
This concept highlights the transformation and conservation of energy, emphasizing that in an isolated system, energy is never lost but merely changes forms. When the marble reaches its maximum height, all kinetic energy initially imparted by the spring has been fully converted to potential energy.

Elastic potential energy is stored in elastic materials as a result of their deformation, such as stretching or compressing a spring. In this context, the energy is stored when the spring is compressed to launch the marble.
The elastic potential energy is calculated using the formula \( \Delta U_s = \frac{1}{2} k x^2\), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position (compression or stretching distance).
This type of potential energy is crucial because it illustrates how energy can be stored in objects temporarily and released to perform work, like propelling the marble upward. Understanding elastic potential energy helps in comprehending energy storage in everyday objects, such as rubber bands, trampolines, and even car suspensions.
In our exercise, the elastic potential energy of the compressed spring becomes kinetic energy that propels the marble, eventually turning into gravitational potential energy at the peak of ascent.

The spring constant, often denoted by \( k \), is a measure of a spring's stiffness. It represents the force required to compress or stretch the spring by a unit length.
A higher spring constant value means a stiffer spring, which requires more force to compress or stretch. Conversely, a lower value indicates a more flexible spring.
In the context of our exercise, the spring constant \( k \) can be derived from the elastic potential energy formula. Here, the marble's kinetic energy (from the spring) and its gravitational potential energy tell us how the spring's energy is stored and used.
Calculating the spring constant is crucial for understanding the amount of potential energy stored and predicting the spring's behavior under different conditions. When a spring gun shoots a marble, the stiffness directly influences how much energy can be transferred from the spring to the marble, ultimately defining how far or high the object can be propelled.