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Gravitational potential energy (U_i) is the energy an object possesses due to its position in a gravitational field. This concept is crucial when an object, like a hammer, is dropped from a height. The energy is stored because of the gravitational attraction between the object and Earth.
The formula for gravitational potential energy at a height \( h \) above the Earth's surface is given by:\[U_i = -\frac{G m m_{\text{E}}}{R_{\text{E}} + h}\]where \( G \) is the gravitational constant, \( m \) is the mass of the hammer, \( m_{\text{E}} \) is the mass of Earth, and \( R_{\text{E}} \) is the radius of Earth.
When the hammer is at height \( h \), it has maximum potential energy. As it falls, this potential energy is converted into kinetic energy as it approaches the Earth. Understanding this energy transformation is key to applying the principle of energy conservation.

Kinetic energy (K) is the energy an object has due to its motion. When an object is in motion, like a hammer falling to the ground, this energy is in play. For falling objects, kinetic energy increases as potential energy decreases.
The formula for kinetic energy is:\[K_f = \frac{1}{2} m v^2\]where \( m \) is the mass of the object, and \( v \) is its velocity.
In the context of a hammer falling from a height, just before it hits the ground, all initial potential energy has been converted to kinetic energy. Understanding this shift is essential for applying energy conservation principles. As the hammer speeds up, its kinetic energy increases, illustrating the inverse relationship between kinetic and potential energy.

The gravitational constant (G) is a fundamental constant in nature, representing the strength of gravity. It is used in Newton's law of universal gravitation and is crucial in calculating gravitational forces and energy.
In mathematical terms, \( G \) is used in equations such as:

• The gravitational force equation \( F = \frac{G m_1 m_2}{r^2} \)
• The potential energy equation \( U = -\frac{G m m_{\text{E}}}{R_{\text{E}} + h} \)

The value of \( G \) is approximately \( 6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2 \). This constant is pivotal in calculating how gravitational forces affect the movement of objects, such as a hammer in free fall.
Understanding \( G \) helps us predict how objects interact in a gravitational field, like how fast a hammer will fall to Earth.

Free fall motion occurs when gravity is the only force acting on an object. In the absence of air resistance, this motion is characterized by a constant acceleration towards the Earth.
With free fall, an object starts at rest and accelerates due to gravity (\( g \)). However, when discussing larger distances or heights like in our scenario, we must consider more precise gravitational influences, including variations at different heights.
Key points about free fall include:

• Acceleration due to gravity is approximately \( 9.81 \text{m/s}^2 \) near the Earth's surface.
• In vacuum conditions, all objects fall with the same acceleration irrespective of their mass.
• The velocity of a freely falling object continuously increases as it falls.

Considering these concepts with our hammer, free fall motion describes how it speeds up as it nears the Earth, transforming potential energy into kinetic energy smoothly and predictably.