# Proving Total Kinetic Energy of Particles

• ougoah
In summary, Total kinetic energy of particles refers to the sum of the individual kinetic energies of all the particles in a system. It is calculated using the formula: KE = 1/2 * m * v^2, where KE is kinetic energy, m is the mass of the particle, and v is the velocity of the particle. The total kinetic energy is affected by the mass and velocity of the particles, as well as external forces such as friction or collisions. Calculating this energy is important in understanding the behavior of a system, and it cannot be negative, although the kinetic energy of individual particles can have negative values in certain cases.
ougoah
For a system of particles, how would you prove that the total kinetic energy of the particles is equal to the kinetic energy associated with the center of mass motion plus the "internal energy"? (Where, I think, internal energy is the energy of the particles seen from the center of mass reference frame.)

Thanks

I don't know why this was moved, since it is not a homework question; my textbook is missing this proof.

for your question! Proving this relationship between total kinetic energy and the kinetic energy associated with the center of mass motion plus the internal energy can be done using basic principles of classical mechanics and the definition of total kinetic energy.

First, let's define the terms in this relationship. The total kinetic energy (KEtotal) is the sum of the kinetic energy of all the particles in the system, while the kinetic energy associated with the center of mass motion (KEcm) is the kinetic energy of the system as a whole, moving with a certain velocity. The internal energy (U) is the energy of the particles relative to the center of mass reference frame.

To prove the relationship, we can start by considering the total kinetic energy of the system. According to the definition of kinetic energy, it is given by the sum of the individual kinetic energies of all the particles in the system:

KEtotal = Σ (1/2)mv²

Next, we can express the velocity of each particle relative to the center of mass reference frame as v' = v - V, where V is the velocity of the center of mass. This allows us to rewrite the total kinetic energy as:

KEtotal = Σ (1/2)m(v' + V)²

Expanding this expression and rearranging terms, we get:

KEtotal = Σ (1/2)mv'² + Σ (1/2)mV² + Σ mVv'

The first term on the right-hand side is the sum of the kinetic energies of the particles relative to the center of mass reference frame, which we can define as the internal energy (U). The second term is simply the kinetic energy associated with the center of mass motion (KEcm). And the third term represents the total momentum of the system, which is equal to the momentum of the center of mass (Momentum = MV) multiplied by the velocity of the center of mass (V). Since the momentum of the center of mass is always conserved, this term will always be equal to zero.

Therefore, we can rewrite the total kinetic energy as:

KEtotal = U + KEcm

This proves the relationship between total kinetic energy and the kinetic energy associated with the center of mass motion plus the internal energy. It shows that the total kinetic energy of a system can be broken down into two components: the kinetic energy associated with the center of mass motion and the internal energy of the particles relative to the center of mass

## 1. What is total kinetic energy of particles?

Total kinetic energy of particles refers to the sum of the individual kinetic energies of all the particles in a system. It is a measure of the overall movement or motion of the particles within a system.

## 2. How is total kinetic energy of particles calculated?

Total kinetic energy of particles can be calculated using the formula: KE = 1/2 * m * v^2, where KE is kinetic energy, m is the mass of the particle, and v is the velocity of the particle.

## 3. What factors affect the total kinetic energy of particles?

The total kinetic energy of particles is affected by the mass and velocity of the particles. An increase in either of these factors will result in an increase in the total kinetic energy. Additionally, external forces such as friction or collisions can also affect the total kinetic energy of particles.

## 4. Why is it important to calculate the total kinetic energy of particles?

Calculating the total kinetic energy of particles is important in understanding the overall behavior and dynamics of a system. It can also help in predicting and analyzing the effects of external forces on the particles.

## 5. Can the total kinetic energy of particles be negative?

No, the total kinetic energy of particles cannot be negative as it is a measure of motion and motion cannot have negative values. However, the kinetic energy of individual particles can be negative if their velocity is in the opposite direction of their motion.

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