Kinetic Energy and Momentum Relations

Fundamental Equations of Kinetic Energy

• The transcript establishes the standard formula for Kinetic Energy ($KE$):
  • $KE = \frac{1}{2} m v^2$
• A standalone expression is mentioned at the start of the notes:
  • $V = \frac{1}{2}$
• The number $2$ is noted as a subscript or standalone value in the initial context.

Proportionality of Kinetic Energy Relative to Mass

• When velocity ($v$) is held as a constant ($v = \text{constant}$):
  • Kinetic Energy ($KE$) is directly proportional to mass ($m$).
  • Relationship: $KE \propto m$
• The ratio for comparing two different states of kinetic energy based on varying mass (at constant velocity) is defined as:
  • $\frac{K_1}{K_2} = \frac{m_1}{m_2}$

Proportionality of Kinetic Energy Relative to Velocity

• When mass ($m$) is held as a constant ($m = \text{constant}$):
  • Kinetic Energy ($KE$) is directly proportional to the square of the velocity ($v^2$).
  • Relationship: $KE \propto v^2$
• The ratios for comparing two different states of kinetic energy based on varying velocity (at constant mass) are defined as:
  • $\frac{K_1}{K_2} = (\frac{v_1}{v_2})^2$
  • Alternatively expressed as: $\frac{K_1}{K_2} = \frac{v_1^2}{v_2^2}$

Relationship Between Kinetic Energy and Momentum

• The notes derive the relationship between Kinetic Energy ($KE$) and Linear Momentum ($P$):
  • The formula is given as: $KE = \frac{P^2}{2m}$
• To solve for momentum ($P$) when Kinetic Energy and mass are known, the formula is rearranged as:
  • $P = \sqrt{2 m KE}$

Proportionality and Ratios of Momentum and Mass

• Scenario 1: Constant Mass

  • When mass ($m$) is constant ($m = \text{constant}$):
    • Kinetic Energy is proportional to the square of momentum: $KE \propto P^2$
  • The ratio for comparing kinetic energies at constant mass with varying momentum is:
    • $\frac{K_1}{K_2} = \frac{P_1^2}{P_2^2}$

• Scenario 2: Constant Momentum

  • When momentum ($P$) is constant ($P^2 = \text{constant}$):
    • Kinetic Energy is inversely proportional to mass: $KE \propto \frac{1}{m}$
  • The ratio for comparing kinetic energies at constant momentum with varying mass is:
    • $\frac{K_1}{K_2} = \frac{m_2}{m_1}$
  • Specific notation provided: $M_2 K_1 = M_1 K_2$, which leads to the inverse ratio sequence.

Kinetic Energy in Terms of Force and Distance

• The transcript provides expressions relating Kinetic Energy to Force ($F$) and a distance or radius component ($r$):
  • $KE = \frac{1}{2} m v^2$
  • Using the substituted relationship: $KE = \frac{1}{2} F \times r$
• Rearranging this formula to solve for Force ($F$):
  • $F = \frac{2 KE}{r}$

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