How Is the Classical Energy-Momentum Relation Derived?

[Erika Martin]

How is derived the classical energy-momentum relation, E = p^2/2m?

Thanks!

 

[lavalamp]

Kinetic Energy = $$0.5 * mv^{2}$$

momentum = p = mv

Kinetic Energy = $$0.5 * p * v$$
Kinetic Energy = $$0.5 * p * p/m$$
Kinetic Energy = $$0.5 * p^{2}/m$$

Kinetic Energy = $$p^{2}/2m$$

 

[Moetasim]

The classical energy-momentum relation, E = p^2/2m, is derived from the principles of classical mechanics, specifically the conservation of energy and momentum. This relation states that the energy of a particle is equal to its momentum squared divided by twice its mass.

To understand how this relation is derived, we must first define the terms involved. Energy is a measure of a system's ability to do work, while momentum is a measure of its motion. In classical mechanics, energy and momentum are both conserved quantities, meaning they remain constant in a closed system. This means that the total energy and momentum of a system before and after any interaction or event must be the same.

The classical energy-momentum relation can be derived mathematically using the equations for kinetic energy (KE) and momentum (p):

KE = 1/2mv^2 and p = mv

Substituting the expression for momentum into the equation for kinetic energy, we get:

KE = 1/2m(mv)^2 = p^2/2m

This shows that the kinetic energy of a particle is directly proportional to its momentum squared and inversely proportional to its mass, which is the classical energy-momentum relation.

One way to interpret this relation is through the concept of work. Work is defined as the force applied to an object multiplied by the distance it moves. In classical mechanics, the work done on an object is equal to its change in kinetic energy. Using this definition, we can see that the classical energy-momentum relation can also be derived from the work-energy theorem, which states that the work done on an object is equal to its change in kinetic energy:

W = KE2 - KE1 = p^2/2m - 0 = p^2/2m

This further reinforces the idea that the classical energy-momentum relation is a result of the conservation of energy and momentum in classical mechanics.

In summary, the classical energy-momentum relation, E = p^2/2m, is derived from the principles of conservation of energy and momentum in classical mechanics. It shows the direct relationship between the energy and momentum of a particle, and can also be derived from the work-energy theorem. This relation is a fundamental concept in classical mechanics and has many applications in understanding the behavior of particles in motion.