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Last day, my cousin sister Natasha, bumped into her brother while running across the room. The impact of the collision pushed the poor chap to the floor. This is one of the daily life instances that define the law of conservation of energy. Wondering how? The kinetic energy possessed by Natasha due to her movement was transferred to her brother, thus, causing him to fall.

In Chemistry and Physics, the law of conservation states that energy can neither be created nor destroyed. Rather, it can only be transferred or transformed from one to another. Initially, the entire concept of conservation of energy was different from conservation of mass. However, special relativity theory proved that mass is related to energy and vice versa. Thus, the formula *E=mc** ^{2}* was established and justified.

Not only that the conservation of energy can be justified by *E=mc** ^{2}*, but it can also be proven by the Noether’s theorem that establishes the fact that the laws of Physics remain unchanged, irrespective of time.

There are various dimensions associated with this particular subject matter. In case, you are willing to explore each one of them, then here’s everything you should know.

A self-adjoint operator named Hamiltonian describes the energy of quantum system. It is said if the Hamiltonian is a time-independent operator, emergence probability of the measurement remains unchanged throughout the evolution of the system. As a result, the expectation of the value of energy tends to remain time-independent as well.

The quantum Noether’s theorem for energy-momentum tensor operator ensures the local energy conservation in Quantum field theory. It is also to be noted, due to the lack of universal time operator in Quantum theory, the relation of uncertainty for energy and time is not fundamental in contrast to the position-momentum uncertainty principle.

Thus, energy in each fixed time in principle can exactly be measured without any trade-off forced by the time-energy uncertainty relations.

You will be required to implement the Schrödinger equation in order to describe quantum mechanical behaviour. The time-dependent one-dimensional Schrödinger equation is established by h^{2}/2m∂^{2}/∂x^{2}Ψ=EΨ, where ? stands for Planck’s constant, m stands for mass, ψ for wave function.

The special relativity theory discovered by Albert Einstein and Henri Poincaré states that energy is proposed to be one of the components of an energy-momentum 4 vector. Each of the four components of energy (three of momentum and one of energy) of this vector is conserved separately across time in any closed system, and as observed from any given inertial reference frame.

In addition to it, the vector length is also conserved, which is determined as the rest mass (the mass of the body when at rest) for single particles, and invariant mass for various systems of particles. It is to be noted, in invariant mass, * momenta and energy* are separately summed prior to calculating the length.

As mentioned and talked about in the initial segment of this blog, Einstein’s theory of Special Relativity (*E = mc** ^{2}*)comes into play in this context. There are three basic principles of the relativity theory. Here’s everything you need to know.

- Objects that are subjected to mobility, or at rest, remain in motion, unless an external force imposes any kind of change in motion.
- Force is equivalent to the change in momentum per change of time. For is equal to the mass time acceleration for a constant Mass.
- For every action, there is an equal and opposite reaction.

The particular relativity theory discovered by Albert Einstein and Henri Poincaré states that energy is proposed to be one of the components of an energy-momentum four-vector. Each of the four elements of energy (three of momentum and one of energy) of this vector is conserved separately across time in any closed system, and as observed from any given inertial reference frame.

In addition to it, the vector length is also conserved, which is determined as the rest mass (the mass of the body when at rest) for single particles, and invariant mass for various systems of particles. It is to be noted, in invariant mass, * momenta and energy* are separately summed before calculating the length.

As mentioned and talked about in the initial segment of this blog, Einstein’s theory of Special Relativity (* E = mc2) *comes into play in this context. There are three basic principles of the relativity theory. Here’s everything you need to know.

- Objects that are subjected to mobility, or at rest, remain in motion unless an external force imposes any change in motion.

- Force is equivalent to the change in momentum per change of time. Force is equal to the mass time acceleration for a constant Mass.

- For every action, there is an equal and opposite reaction.

In Physics, the term conservation refers to something that remains constant. This indicates the fact that the variable in an equation which represents a conserved quantity is continuous over time. It will have the same value, both before and after an event. It is to be noted that there are various conserved quantities in physics. They are often considered useful for making predictions.

Talking of the law of conservation of energy, it is to be mentioned that there are three fundamental quantities. These are angular momentum, momentum and energy.

:**Angular Momentum**

The angular momentum of a concrete object is referred to as the product of the moment of inertia and angular velocity. It is to be noted that angular momentum is analogous to linear momentum. Also, it is subjected to the fundamental constraints of the conservation of angular momentum principle, provided there is no external torque applied on the object.

:**Momentum**

Momentum is defined as the quantity of motion present in an object. Any object which is motion has momentum, and can be referred to as “mass in motion”. Momentum depends on two key variables; Mass and Velocity.

:**Energy**

Goes without saying, energy can be found in many things, and many shapes and forms. It can be further categorised as renewable and non-renewable depending on the type or nature of the source.

When energy is conserved we can establish equations that will equate the sum of the different types of energy in the system. Once the equation is set up, we can gradually move on the aspects of solving equations for velocity, distance and any other parameter on which energy depends. In case we are not able to solve the equations for velocity or we don’t have enough variables to find a unique solution, then it could still be useful to plot related variables to observe where the solution lies.

Here is a famous example - if we are to consider a golfer striking a golf ball with gravitational acceleration 1.625 m/s^22. The ball leaves at an angle of 45 degree to the lunar surface travelling at 20 m/s, both vertically and horizontally, with a total velocity of 28.28 m/s.

Now, if we are to find out how high the golf ball would go, the equation * E*M=21

By implementing the principle of conservation of mechanical energy, we can find out the value of h (height) that is 123m.

**Parting Thoughts**

I hope this informative blog on the different fundamentals of the ** law of conservation** of energy will help you develop a thorough insight into the nitty-gritty details. Take note of the key points and aspects highlighted in this write-up and venture out to compose a winning assignment on this topic like a pro.

Good luck!

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