Derivation of relativistic energy momentum equation (total energy) • HERO OF THE DERIVATIONS.

Summary

In this video, the concept of total energy for subatomic particles like protons and electrons is discussed, with a focus on Einstein's mass-energy equivalence formula. The script covers the relationship between rest mass, moving mass (gamma), and momentum, introducing key equations that describe particle energy. The importance of the Lorentz factor, gamma, is highlighted in calculating energy for moving particles. The video also touches on the special case of photons, where the mass is considered zero, resulting in a simplified energy expression equal to momentum times the speed of light.

Takeaways

• 😀 The energy associated with a particle is explained using Einstein's mass-energy equivalence formula: E = mc².
• 😀 The mass in motion (relativistic mass) is given by the formula: m' = m / √(1 - v²/c²), where m is the rest mass, v is velocity, and c is the speed of light.
• 😀 The Lorentz factor (γ) is defined as γ = 1 / √(1 - v²/c²), which adjusts the mass and energy for objects in motion.
• 😀 The momentum of a moving particle is given by p = m'v, where m' is the relativistic mass and v is the velocity.
• 😀 The total energy of a moving particle is calculated as E = γmc², where γ is the Lorentz factor and m is the rest mass.
• 😀 The energy equation for a moving particle is squared to give the relativistic energy formula: E² = m²c⁴ + p²c².
• 😀 The derived formula E² = m²c⁴ + p²c² highlights the relationship between rest energy, kinetic energy, and momentum for a particle.
• 😀 For a photon (massless particle), the energy simplifies to E = pc, where p is the momentum and c is the speed of light.
• 😀 A photon has no rest mass (m = 0), so its energy is solely determined by its momentum, making E = pc a crucial equation for photons.
• 😀 The script emphasizes the importance of understanding relativistic effects when dealing with particles moving at speeds close to the speed of light.

Q & A

What is the mass-energy equivalence formula?

The mass-energy equivalence formula is given by E = mc², where E is the energy, m is the rest mass of the particle, and c is the speed of light.

What does the term 'rest mass' refer to?

Rest mass refers to the mass of a particle when it is at rest, meaning its velocity is zero.

What happens to the mass of a particle when it is in motion?

When a particle is in motion, its mass increases according to the equation M' = M / √(1 - v²/c²), where M is the rest mass, v is the velocity of the particle, and c is the speed of light.

What is the 'Lorentz Factor'?

The Lorentz Factor, denoted by gamma (γ), is given by γ = 1 / √(1 - v²/c²). It accounts for the increase in mass and the time dilation experienced by a moving object.

How is the momentum of a particle expressed when it is in motion?

The momentum of a particle in motion is given by p = M'v, where M' is the relativistic mass, and v is the velocity of the particle.

What is the relationship between velocity, momentum, and mass?

Velocity is related to momentum and mass through the equation v = p / (γM), where p is the momentum, γ is the Lorentz factor, and M is the rest mass of the particle.

What is the significance of the equation for gamma (γ)?

The equation for gamma (γ) allows us to calculate the relativistic mass when an object is in motion, adjusting the rest mass for the effects of speed approaching the speed of light.

How does the total energy of a moving particle differ from the energy of a stationary particle?

The total energy of a moving particle is given by E = γMc², where γ accounts for the motion of the particle, while for a stationary particle, the energy is simply E = mc².

What happens when you square the energy equation?

Squaring the energy equation E = γMc² results in E² = γ²M²c⁴, which can then be simplified further using the momentum equation to yield E² = M²c⁴ + p²c².

What is the total energy of a photon?

The total energy of a photon is given by E = pc, where p is the momentum of the photon and c is the speed of light. Since the photon has no rest mass, the rest mass term disappears in its energy equation.