Introduction
Conservation of momentum and energy require relativistic modifications. Momentum becomes p = γmv, energy E = γmc2. These reduce to Newtonian forms at low velocities but ensure conservation laws hold at all speeds. Mass-energy equivalence E = mc2 emerges naturally. These are fundamental for particle physics.
Relativistic Momentum
p = γmv = mv/√(1-v2/c2). At v << c: p ≈ mv (Newtonian). As v → c: p → ∞, requiring infinite force to reach c. Conserved in isolated systems. Direction is parallel to velocity as in Newtonian mechanics. Momentum conservation holds in all inertial frames.
Relativistic Energy
Total energy: E = γmc2 = mc2/√(1-v2/c2). At v = 0: E = mc2 (rest energy). At v << c: E ≈ mc2 + 1/2mv2 (rest + kinetic). Kinetic energy: K = (γ-1)mc2 = E - mc2. K → ∞ as v → c. No upper limit to kinetic energy, but velocity limited to c.
\nIntroduction
Conservation of momentum and energy require relativistic modifications. Momentum becomes p = γmv, energy E = γmc2. These reduce to Newtonian forms at low velocities but ensure conservation laws hold at all speeds. Mass-energy equivalence E = mc2 emerges naturally. These are fundamental for particle physics.
Relativistic Momentum
p = γmv = mv/√(1-v2/c2). At v << c: p ≈ mv (Newtonian). As v → c: p → ∞, requiring infinite force to reach c. Conserved in isolated systems. Direction is parallel to velocity as in Newtonian mechanics. Momentum conservation holds in all inertial frames.
Relativistic Energy
Total energy: E = γmc2 = mc2/√(1-v2/c2). At v = 0: E = mc2 (rest energy). At v << c: E ≈ mc2 + 1/2mv2 (rest + kinetic). Kinetic energy: K = (γ-1)mc2 = E - mc2. K → ∞ as v → c. No upper limit to kinetic energy, but velocity limited to c.
\nEnergy-Momentum Relation
E2 = (pc)2 + (mc2)2. Invariant relation connecting energy, momentum, and mass. For massless particles (light): E = pc. For massive particles at rest: E = mc2. Hyperbolic relation in E-p space. All observers agree on mass (rest energy), but E and p vary with frame.
Mass-Energy Equivalence
E = mc2: mass is a form of energy, convertible to other forms. Nuclear reactions: mass difference becomes kinetic energy (fission, fusion). Binding energy: bound systems have less mass than constituents. Validates conservation of total mass-energy. Most famous equation in physics.
Applications
Particle physics: accelerators require relativistic formulas for momentum and energy. Nuclear energy: mass-energy conversion in reactors and bombs. Pair production: photon energy creates particle-antiparticle pairs (E = 2mc2 needed). Annihilation: matter-antimatter converts entirely to energy. PET scans use annihilation photons.
\nSolved Example: Relativistic Electron
An electron (m = 9.11×10⻳¹ kg = 0.511 MeV/c2) is accelerated to kinetic energy K = 10 MeV. Find: (a) Total energy, (b) Lorentz factor, (c) Velocity, (d) Momentum. Solution: Rest energy E0 = mc2 = 0.511 MeV. (a) Total energy E = E0 + K = 0.511 + 10 = 10.511 MeV. (b) γ = E/E0 = 10.511/0.511 = 20.57. (c) From γ = 1/√(1-v2/c2): 1/γ2 = 1 - v2/c2 → v2/c2 = 1 - 1/γ2 = 1 - 1/422.9 = 0.9976. v = 0.9988c ≈ 0.999c. Very close to c but not exceeding it. (d) Momentum: p = √(E2 - E02)/c = √(110.5 - 0.261)/c MeV = √110.2 MeV/c = 10.5 MeV/c. Or p = γmv = 20.57 × 9.11×10⻳¹ × 0.9988 × 3×10^8 kg·m/s = 5.61×10⻲¹ kg·m/s. Converting: 1 MeV/c = 5.34×10⻲2 kg·m/s, so p = 10.5 × 5.34×10⻲2 = 5.61×10⻲¹ kg·m/s ✓. For comparison, Newtonian momentum would be mv = 9.11×10⻳¹ × 0.9988 × 3×10^8 = 2.73×10⻲2 kg·m/s, about 20× smaller than relativistic momentum. At high energies, relativistic momentum dominates.
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