Introduction
Total linear momentum P of an isolated system is conserved when net external force is zero. This powerful principle applies to collisions, explosions, and isolated systems. Momentum conservation is more fundamental than Newton's laws, remaining valid even in quantum mechanics and relativity where classical mechanics breaks down.
Statement of Conservation
If the net external force on a system is zero (F_ext = 0), then the total linear momentum P = Σ p_i = Σ m_i v_i is constant: dP/dt = 0, so P = constant. This means P_initial = P_final at any two times. The conservation holds separately for each component: if F_x = 0, then P_x is conserved, even if other components change.
Conditions for Conservation
Momentum is conserved when: (1) Net external force is exactly zero; (2) External forces are negligible compared to internal forces during brief interactions (collisions, explosions); (3) A specific component of external force is zero, conserving that momentum component. Internal forces (action-reaction pairs between system parts) never change total momentum - they cancel by Newton's Third Law.
\nIntroduction
Total linear momentum P of an isolated system is conserved when net external force is zero. This powerful principle applies to collisions, explosions, and isolated systems. Momentum conservation is more fundamental than Newton's laws, remaining valid even in quantum mechanics and relativity where classical mechanics breaks down.
Statement of Conservation
If the net external force on a system is zero (F_ext = 0), then the total linear momentum P = Σ p_i = Σ m_i v_i is constant: dP/dt = 0, so P = constant. This means P_initial = P_final at any two times. The conservation holds separately for each component: if F_x = 0, then P_x is conserved, even if other components change.
Conditions for Conservation
Momentum is conserved when: (1) Net external force is exactly zero; (2) External forces are negligible compared to internal forces during brief interactions (collisions, explosions); (3) A specific component of external force is zero, conserving that momentum component. Internal forces (action-reaction pairs between system parts) never change total momentum - they cancel by Newton's Third Law.
\nCenter of Mass Motion
Total momentum equals total mass times center of mass velocity: P = M V_CM. When P is conserved, V_CM is constant. If system starts at rest, CM remains at rest regardless of internal motions. If system has initial CM velocity, CM continues with that velocity. This explains why a freely exploding projectile's fragments have CM that follows original parabolic path.
Applications to Collisions
In collisions, momentum is always conserved (during the brief collision, external forces are negligible compared to impact forces). For two bodies: m1v1 + m2v2 = m1v1' + m2v2' before and after. This applies to: car crashes, ball sports, atomic scattering, rocket propulsion. Combined with energy considerations (elastic vs inelastic), this determines post-collision motion completely.
\nRockets and Propulsion
Rocket propulsion demonstrates momentum conservation: initially stationary rocket+fuel has zero momentum. Expelling exhaust backward at high speed requires rocket to move forward so total momentum remains zero (in rocket's initial rest frame). Thrust arises from rate of momentum transfer to exhaust. This works in vacuum - no need to push against external medium.
Solved Example: Cannon Recoil
A 500 kg cannon fires a 5 kg projectile at 200 m/s horizontally. Find recoil velocity of cannon and compare kinetic energies. Solution: Initial momentum = 0. By conservation: 0 = m_projectile × v_p + m_cannon × v_c → v_c = -m_p × v_p / m_c = -5 × 200 / 500 = -2 m/s. Cannon recoils at 2 m/s opposite to projectile. KE of projectile: 1/2 × 5 × (200)2 = 100,000 J. KE of cannon: 1/2 × 500 × (2)2 = 1,000 J. Despite equal momentum magnitudes, projectile has 100× more kinetic energy because KE = p2/(2m) and m_p << m_c. This is why projectiles do damage while recoil is manageable.
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