Momentum

Momentum is a measure of the motion of an object, defined as the product of its mass and its velocity: p = m v. It is a vector quantity, with magnitude |p| = m|v| and direction along the velocity vector. The unit is kilogram-meter per second (kg·m/s).

In classical mechanics, momentum is conserved in any closed system where external forces are negligible. The total momentum before an interaction equals the total after. The impulse imparted on an object equals the change in its momentum: Δp = F Δt, or F = dp/dt.

In collisions, momentum is conserved even if kinetic energy is not (inelastic collisions). In elastic collisions, both momentum and kinetic energy are conserved; in perfectly inelastic collisions, objects stick together and share a common velocity.

The total momentum of a system relates to its center of mass: P = M V_CM, where M is the total mass and V_CM is the velocity of the center of mass. In the absence of external forces, the center of mass moves with constant velocity.

Relativistic momentum generalizes p = m v to p = γ m v, where γ is the Lorentz factor. The energy–momentum relation becomes E^2 = (pc)^2 + (mc^2)^2, linking momentum to total energy at high speeds.

In quantum mechanics, momentum is associated with the wave nature of matter; the momentum operator is -iħ ∇, and de Broglie’s relation p = h/λ connects momentum with wavelength.

Applications of momentum span propulsion, collisions, and fluid dynamics, where momentum conservation and momentum flux play central roles in both theory and experiment.