State and prove law of conservation of energy in case of a freely falling body?

The law of conservation of energy is a fundamental principle in physics that states that energy cannot be created or destroyed; it can only be transformed from one form to another. In the context of a freely falling body, this law can be illustrated beautifully through the interplay of gravitational potential energy and kinetic energy as the body falls. Let’s break this down step by step.

Understanding the Concepts

When an object is at rest at a height above the ground, it possesses gravitational potential energy (PE) due to its position in a gravitational field. As the object begins to fall, this potential energy is converted into kinetic energy (KE), which is the energy of motion. The total mechanical energy (the sum of potential and kinetic energy) of the object remains constant throughout its fall, assuming no air resistance.

Mathematical Representation

We can express the energies involved using formulas:

• Potential Energy (PE): PE = mgh
• Kinetic Energy (KE): KE = 0.5mv²

Here, m is the mass of the body, g is the acceleration due to gravity (approximately 9.81 m/s² on Earth), h is the height above the ground, and v is the velocity of the body.

Proving the Law of Conservation of Energy

Let’s consider a body of mass m that is dropped from a height h. Initially, when the body is at rest at height h, its potential energy is:

Initial PE = mgh

As the body falls, it accelerates due to gravity. At a height h' (where h' < h), the potential energy becomes:

PE at height h' = mgh'

Simultaneously, the body gains kinetic energy as it falls. The velocity of the body at height h' can be found using the equation of motion:

v² = u² + 2gh (where u is the initial velocity, which is 0)

This simplifies to:

v² = 2g(h - h')

Now, substituting this into the kinetic energy formula:

KE at height h' = 0.5m(2g(h - h')) = mg(h - h')

Energy Conservation Equation

According to the law of conservation of energy, the total mechanical energy at the initial height should equal the total mechanical energy at height h':

Initial PE = PE at h' + KE at h'

Substituting the values we derived:

mgh = mgh' + mg(h - h')

When we simplify the right side:

mgh = mgh' + mgh - mgh'

This simplifies to:

mgh = mgh

This equality holds true, demonstrating that the total mechanical energy remains constant throughout the fall. Thus, the law of conservation of energy is proven in the case of a freely falling body.

Real-World Implications

This principle is not only fundamental in physics but also has practical applications in various fields, such as engineering, environmental science, and even in understanding celestial mechanics. For example, when designing roller coasters, engineers must consider the conversion of potential energy at the highest points into kinetic energy as the cars descend, ensuring safety and thrill.

In summary, the law of conservation of energy in the case of a freely falling body illustrates the seamless transformation between potential and kinetic energy, reinforcing the idea that energy is a constant in our universe, merely changing forms as it interacts with forces like gravity.