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PhysicsDifficulty 1-3

Forces

A push or pull interaction between two objects, capable of changing an object's velocity. Newton's three laws of motion describe how forces and motion relate: a body keeps its state of motion unless acted on by a net force (first law), net force equals mass times acceleration, F = ma (second law), and forces between two bodies are equal and opposite (third law).

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Newton's Second Law in General Form

F = ma is the constant-mass special case of a more general statement: net force is the rate of change of momentum.

F=dpdt=d(mv)dt\mathbf{F} = \dfrac{d\mathbf{p}}{dt} = \dfrac{d(m\mathbf{v})}{dt}

Newton's second law in terms of momentum p = mv.

When mass is constant, the product rule reduces this to the familiar F = ma. The momentum-based form is more general: it still applies to systems where mass itself changes over time, such as a rocket burning and ejecting fuel, where F = ma alone is not sufficient.

F=mdvdt+vdmdt\mathbf{F} = m\dfrac{d\mathbf{v}}{dt} + \mathbf{v}\dfrac{dm}{dt}

Expanding d(mv)/dt by the product rule shows the extra term that appears when mass varies with time.

Info: This is exactly the relationship connecting Forces to Momentum: an impulse — a force applied over a time interval — is the integral of force with respect to time, and it equals the resulting change in momentum.

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Sources and methodology

  • Newton's first law of motion states that an object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted on by a net external force.
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • Newton's second law of motion states that the net force on an object equals its mass multiplied by its acceleration (F = ma).
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • Newton's third law of motion states that for every force one object exerts on a second object, the second object exerts an equal and opposite force back on the first.
    • Fundamentals of Physics (Halliday, Resnick & Walker)

Content status: published, last reviewed 30 September 2026.