Titus 2/26

Momentum Principle

• Key Concept: The momentum principle is foundational, related to the conservation of momentum.

• Application: This principle has been emphasized for the first two tests in the semester.

• Conservation of Momentum: Important for understanding the relationships and behaviors in physics.

Energy Principle

• Introduction: The next crucial principle is the energy principle.

• Key Definition: The change in the energy of a system equals the sum of energy inputs from the surroundings.

  • Positive inputs increase energy.

  • Negative inputs can be viewed as outputs that decrease energy.

• Analogy: The principle can be likened to a bank account where the change in balance reflects both inputs and outputs.

Types of Energy in Particles

• Basic Definition: A particle is an idealized entity used for simplifying complex systems.

• Rest Energy: Energy associated with a particle's mass.

  • Famous Equation: The rest energy is represented by E = mc², where 'c' is the speed of light.

• Kinetic Energy: Associated with the motion of a particle.

  • Total energy (E) comprises rest energy and kinetic energy.

  • Total energy formula: E = γmc², where γ = 1/sqrt(1 - v²/c²).

Calculating Kinetic Energy

• Expression for Kinetic Energy: Kinetic energy can also be expressed as:

  • KE = (γ - 1)mc².

• Low Speed Approximation: For objects moving below 10% of the speed of light, the kinetic energy is approximately:

  • KE ≈ 1/2 mv².

  • Derived from the series expansion of γ for low speeds.

Practice Problems

• Example with a Ball:

  • Ball mass: 2 kg.

  • Check if speed is under 10% of the speed of light to use right formula.

  • Kinetic energy calculation: KE = (1/2)mv².

  • Important note: Kinetic energy is scalar, not vector.

• Rest Energy Calculation: Use E = mc²; appreciate the significant difference between rest energy and kinetic energy.

  • Example shows that rest energy can be significantly larger than kinetic energy (1.8 x 10¹⁷ J rest energy vs 25 J kinetic energy).

Energy Conversion and Implications

• Mass-Energy Relationship: The substantial rest energy indicates the potential of utilizing mass for energy conversion in nuclear fusion or other processes.