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Gravitational Potential Energy is the energy stored in an object due to its position in a gravitational field. When you lift an object off the ground, you give it gravitational potential energy. The energy is determined by the object's mass, the height it is raised, and the strength of the gravitational field. The formula for gravitational potential energy is \[U_g = mgh\],where:

• \(U_g\) is the gravitational potential energy.
• \(m\) is the mass of the object in kilograms.
• \(g\) is the acceleration due to gravity, approximately \(9.8 \, \text{m/s}^2\) on Earth.
• \(h\) is the height in meters.

This energy converts to other forms when the object moves, such as when a block falls toward a spring. As it falls, gravitational potential energy decreases, transforming into other energy forms like kinetic energy or elastic potential energy once it hits and compresses a spring.
Understanding this concept helps in analyzing how energies transform in physical systems, like in this case, where the energy at height becomes the energy that compresses a spring.

Elastic Potential Energy is the energy stored in elastic materials as a result of their stretching or compressing. When an object like a spring is compressed or stretched, it stores energy that can be released later. The energy depends on both the amount of stretch/compression and the spring’s stiffness, known as the spring constant.
The formula for calculating elastic potential energy is:\[U_e = \frac{1}{2} k x^2\]where:

• \(U_e\) is the elastic potential energy.
• \(k\) is the spring constant, which indicates how stiff the spring is.
• \(x\) is the displacement from the equilibrium position.

In our exercise, when the block hits the spring and compresses it, the gravitational potential energy converts into elastic potential energy. This stored energy is what slows down the block, bringing it momentarily to rest. Understanding this concept allows one to see how systems store and release energy, such as springs in clocks or shock absorbers in vehicles.

The Spring Constant is a measure of a spring's stiffness. It provides an indication of how much force you need to apply to compress or stretch the spring by a certain amount. Denoted by \(k\), it is expressed in units of Newtons per meter (\(\text{N/m}\)).
In Hooke's Law, the spring constant relates the force exerted on the spring and the displacement it causes:\[F = kx\]where:

• \(F\) is the force applied to the spring.
• \(k\) is the spring constant.
• \(x\) is the displacement of the spring from its initial position.

A larger spring constant means a stiffer spring, requiring more force for the same amount of displacement. For instance, in this exercise, a spring constant of 450 N/m means the spring is relatively stiff, requiring a significant force for compression. Understanding the spring constant is crucial for calculating elastic potential energy and interpreting how physical systems behave, especially in cases involving energy transformations like the one in our block and spring scenario.