Newton’s Second Law: Two Equivalent Forms

 Standard Form of Newton’s Second Law: \( \mathbf{F} = m\mathbf{a} \)

This is the most familiar form used in elementary physics. It expresses that the net force \( \mathbf{F} \) acting on an object is equal to its mass \( m \) multiplied by its acceleration \( \mathbf{a} \). This form assumes:

• The mass of the object is constant over time.
• It applies best to simple systems like blocks, cars, or projectiles where mass does not change.
• Acceleration is the direct consequence of the applied force when mass remains fixed.

General (Momentum-Based) Form: \( \mathbf{F} = \dfrac{d\mathbf{p}}{dt} = \dfrac{d}{dt}(m\mathbf{v}) \)

This more fundamental form of Newton’s Second Law expresses force as the time rate of change of momentum \( \mathbf{p} \). It is universally valid, even when mass varies. This form:

• Accounts for both changes in velocity and mass.
• Expands to:
  

  \( \mathbf{F} = \dfrac{d}{dt}(m\mathbf{v}) = m\dfrac{d\mathbf{v}}{dt} + \mathbf{v}\dfrac{dm}{dt} \)

• The second term \( \mathbf{v}\dfrac{dm}{dt} \) becomes significant in systems where mass is not conserved (e.g., rockets, fluid jets, fuel tanks).

 When Are They Equivalent?

When mass is constant, the general momentum form simplifies to the standard form:

\( \dfrac{d}{dt}(m\mathbf{v}) = m\dfrac{d\mathbf{v}}{dt} = m\mathbf{a} \)

So: \( \mathbf{F} = m\mathbf{a} \)

When Is the General Form Needed?

In real-world applications where the mass of the system is changing during motion:

• Rocket propulsion – As fuel burns, the rocket’s mass decreases.
• Leaking tanks or carts losing sand – Mass reduces as material exits the system.
• Collisions involving fragmentation – Total system mass may redistribute dynamically.