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PhysicsDifficulty 3-4

Momentum

The product of an object's mass and velocity, a vector quantity describing the 'quantity of motion' it carries. In a closed system with no external forces, total momentum before and after an interaction such as a collision is conserved, even when kinetic energy is not (as in an inelastic collision).

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Impulse as an Integral of Force

J = FΔt assumes a constant force. Real collision forces typically rise sharply, peak, and fall away again within milliseconds, so the impulse is more generally the integral of force over the time it acts.

J=∫t1t2F dt=Δp\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\,dt = \Delta \mathbf{p}

Impulse as the time-integral of force, equal to the resulting change in momentum.

This is the same relationship as Newton's second law in momentum form, F = dp/dt, integrated over the interval during which the force acts — impulse and force are related to each other exactly as displacement and velocity are, or velocity and acceleration.

Info: Because the integral only depends on the area under the force–time curve, two very different collisions — a short, sharp force and a longer, gentler one — can deliver the identical impulse, and therefore the identical change in momentum, provided the areas under their respective force–time graphs match.

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Sources and methodology

  • Momentum is the product of an object's mass and velocity, a vector quantity (p = mv).
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • In a closed system with no external forces, the total momentum before and after an interaction such as a collision is conserved.
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • An impulse — a force applied over a time interval — equals the change in momentum (J = FΔt = Δp).
    • Fundamentals of Physics (Halliday, Resnick & Walker)
  • Momentum conservation holds even in collisions where kinetic energy is not conserved, such as an inelastic collision.
    • Fundamentals of Physics (Halliday, Resnick & Walker)

Content status: published, last reviewed 30 September 2026.