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Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It depends on three key factors: the mass of the object, the gravitational force acting on it, and the height from which it is dropped. In mathematical terms, it is expressed by the formula:

• \( PE = mgh \)

where:

• \( m \) 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 from which the object is dropped.

In the given exercise, understanding gravitational potential energy is critical as it is the initial form of energy the block possesses. When the block drops, this energy transforms completely into another form of energy when it impacts the spring. This is exactly what makes potential energy a dynamic component in the context of energy conservation, serving as a measurable bridge in the study of motion and force.

Elastic potential energy is the potential energy stored in elastic materials, such as springs, when they are stretched or compressed. The formula for elastic potential energy in a spring is:

• \( ext{Elastic Potential Energy} = \frac{1}{2} k x^2 \)

where:

• \( k \) is the spring constant in newtons per meter (N/m), indicating the spring's stiffness,
• \( x \) is the displacement or compression of the spring from its equilibrium position in meters.

When the block lands on the spring, its gravitational potential energy converts into elastic potential energy, causing the spring to compress. The amount of energy stored is proportional to the square of the compression distance, making the relationship quadratic. This means a small increase in compression leads to a larger increase in elastic potential energy, demonstrating how energy conservation principles apply in practical scenarios, such as cushioning systems, car suspensions, and more.

Spring compression refers to the degree to which a spring is compacted by an external force. It’s a measure of the energy transformation from gravitational to elastic potential energy. In this exercise, the fundamental goal is to determine the maximum compression of the spring when the block lands on it.

• Using energy conservation, the gravitational potential energy of the block is fully transformed into the elastic potential energy of the spring.
• The conversion allows us to solve for compression with the formula relating the energies: \( 9.8 = \frac{1}{2} \times 1960 \times x^2 \).
• By solving the equation, we can derive the compression distance: \( x = \sqrt{0.01} = 0.1 \, \text{meters} \).

This compression calculation is not just a mathematical exercise, but also provides insights into how springs are designed to absorb impact, efficiently converting motion and potential energy into a form that slows down and eventually stops moving objects safely.