Conservation of Momentum and Newton’s Second Law Forms

Newton’s Second Law can be written in two equivalent but context-specific forms:

• Form 1 (constant mass): \( F = ma \)
• Form 2 (changing or constant mass): \( F = \frac{\Delta p}{\Delta t} \)

1. \( F = ma \): The Standard Form (Constant Mass Only)

• Derived from Newton’s second law assuming mass is constant.
• Since momentum is \( p = mv \), differentiating gives:

\( \frac{dp}{dt} = \frac{d(mv)}{dt} = m \frac{dv}{dt} = ma \)

• This is applicable for:
  • Rigid bodies where the mass doesn’t change.
  • Everyday forces on cars, balls, blocks, etc.

2. \( F = \frac{\Delta p}{\Delta t} \): General Form (Mass May Vary)

• This is the more general form of Newton’s Second Law.
• It works even if mass is changing, as in:
  • Rockets losing mass as fuel burns
  • Rain accumulating in a moving cart
  • Jet propulsion
• Momentum is defined as \( p = mv \). The total force is the rate of change of momentum:

\( F = \frac{\Delta p}{\Delta t} = \frac{d(mv)}{dt} = m \frac{dv}{dt} + v \frac{dm}{dt} \)

• This accounts for cases where mass changes over time and can even include variable ejection speeds (as in a rocket).

3. Momentum Conservation Principle

• If there is no external force acting on a system, then from:

\( F = \frac{\Delta p}{\Delta t} \Rightarrow 0 = \frac{\Delta p}{\Delta t} \Rightarrow \Delta p = 0 \)

• This implies that momentum remains constant:

\( p_{\text{initial}} = p_{\text{final}} \)

• Momentum conservation is valid in all types of interactions (e.g., explosions, collisions), provided there’s no net external force.