10–1 Newton's Third Law
Interaction forces come in pairs: equal in magnitude, opposite in direction, and collinear (approximately). Combined with the second law, this leads to conservation of momentum.
10–2 Conservation of Momentum Theorem
dp_1/dt = −dp_2/dt ⇒ d(p_1+p_2)/dt=0
Internal forces do not change the total momentum of a system; without external forces, Σ m_i v_i is constant. Conservation holds independently for each component.
Collisions and Intuitive Derivation from Relativity
- Equal-mass explosion separation: ±v; equal-mass head-on collision and sticking: rest.
- One-dimensional asymmetric: m and 2m stick together: v → v/3; general result mv_1+Mv_2 = (m+M)v'.
- Galilean relativity: uniformly moving reference frames are equivalent; "riding along to observe" simplifies analysis.
10–4 Momentum and Energy
Elastic collision: both momentum and kinetic energy are conserved; inelastic: only momentum is conserved, some kinetic energy converts to internal energy/heat.
Equal-mass elastic collision in 1D: velocities are exchanged; molecular collisions are approximately perfectly elastic.
Rocket/recoil: ejecting small mass m at high velocity V, the main body M recoils with v = (m/M)V.
10–5 Relativistic and Quantum Corrections
- Relativistic momentum: p = m_0 v / √(1−v²/c²), still conserved.
- Field momentum: the electromagnetic field carries momentum and energy; radiation pressure is measurable, ensuring total conservation.
- Quantum mechanics: the definition of momentum changes under wave-particle duality, but the conservation law still holds.
Key Points
- Third law + second law ⇒ total momentum of a system is conserved (no external forces).
- Sticking/explosion/elastic collisions can all be quickly solved using conservation laws.
- Conservation laws hold in both relativistic and quantum theories, though the definition of momentum is updated.