totalmomentum

Total momentum is a fundamental quantity in physics that describes the overall motion of a system. It is the vector sum of the momenta of all constituents. In a system of N particles, P = Σ_i p_i, where p_i = m_i v_i is the momentum of particle i. In continuous media, this becomes P = ∫ ρ(r) v(r) dV, integrating over the system’s volume.

A key property of total momentum is its conservation in isolated systems. If no external forces act on the system, dP/dt = 0, meaning total momentum remains constant. When external forces F_ext are present, the rate of change of total momentum equals the net external force: dP/dt = F_ext. The total momentum is closely tied to the center of mass, with the center-of-mass velocity given by V_cm = P/M, where M = Σ_i m_i is the total mass. In the center-of-mass frame, the total momentum is zero.

Momentum is a vector quantity with direction corresponding to the motion of the system, and its SI unit is kilogram meter per second (kg·m/s). In collisions within an isolated system, total momentum is conserved even though kinetic energy may change; elastic collisions conserve both momentum and kinetic energy, while inelastic collisions conserve momentum but not necessarily kinetic energy.

In relativity, momentum is part of the four-momentum P^μ = (E/c, p), and its invariant length P^μ P_μ = -m^2 c^2 relates momentum to rest mass. In quantum mechanics, momentum becomes an operator, and total momentum is the sum of single-particle momentum operators, linked to translation symmetry via Noether’s theorem.

Applications of total momentum include propulsion analysis, collision studies, and fluid dynamics, where momentum balance and flux are central concepts.