Newton's second law F = ma was derived from classical mechanics where mass is constant. Einstein's Special Theory of Relativity (1905) showed that mass is NOT constant — it increases with velocity. The correct relativistic relation is: m = m₀/√(1−v²/c²) where m₀ is the rest mass (mass when the object is at rest) and c is the speed of light (≈ 3×10⁸ m/s). As v → c, the denominator → 0, so m → ∞. This means infinite force would be needed to accelerate an object to the speed of light — explaining why no material object can reach c. Newton's second law is thus an approximation valid only when v << c.

• The Lorentz factor γ = 1/√(1−v²/c²) ≥ 1 for all velocities (equals 1 at v = 0; increases without bound as v → c). All relativistic quantities are expressed in terms of γ: m = γm₀, p = γm₀v, E = γm₀c², L = L₀/γ, t = γt₀. γ is the 'correction factor' that classical mechanics ignores. For everyday motions (even airplane at 900 km/h = 250 m/s): v/c ≈ 250/(3×10⁸) ≈ 10⁻⁶ — negligible. γ ≈ 1 to 1 part in 10¹². Classical mechanics is perfectly adequate.
• The modified Newton's second law: F = dp/dt where p = γm₀v is the relativistic momentum. At low speeds (v << c): γ ≈ 1, p ≈ m₀v, so F = d(m₀v)/dt = m₀(dv/dt) = m₀a. Classical law recovered. At high speeds: dp/dt has two contributions — the mass−acceleration term AND the velocity×(dm/dt) term. Expanding: F = m(dv/dt) + v(dm/dt). Both terms matter at relativistic speeds.
• Experimental evidence for relativistic mass: (1) Particle accelerators — protons in the LHC are accelerated to 0.9999999c. At this speed γ ≈ 7461, so m ≈ 7461×m_proton. It takes enormously more energy to increase velocity slightly (the 'wall' near c). (2) Muon decay — muons created in the upper atmosphere by cosmic rays travel far enough to reach the Earth's surface. In their rest frame, they should decay before reaching Earth. But time dilation (their 'clock runs slow' relative to Earth) allows them to survive. (3) Mass-energy equivalence E = mc² confirmed in nuclear reactions.

Two equally important consequences of Special Relativity, both following from γ: (1) Length contraction: an object moving at speed v appears (to a stationary observer) to be contracted in the direction of motion: L = L₀√(1−v²/c²) = L₀/γ. The object's proper length L₀ (measured in its rest frame) shrinks by factor γ when measured from a frame in which it moves. (2) Time dilation: a clock moving at speed v appears (to a stationary observer) to run slow: time interval t = t₀/√(1−v²/c²) = γt₀ where t₀ is the proper time (time in the rest frame of the clock). Moving clocks run slow — a moving person ages more slowly relative to a stationary observer.

• Length contraction is only in the direction of motion. Dimensions perpendicular to the motion are unaffected. A moving sphere remains a sphere — wait, it actually becomes an oblate shape because the direction of motion contracts while perpendicular dimensions remain the same. For NEET: a rod moving along its length contracts to L = L₀/γ. A rod moving perpendicular to its length has unchanged length. Key test: direction matters.
• Time dilation: t_moving = γ × t_rest. If t₀ = 1 second on a moving clock, a stationary observer measures γ seconds. The twin paradox: one twin travels at near-c, returns to find the other twin aged more — the travelling twin's clock ran slower. Muon experiment: the muon's lifetime is dilated from the Earth's perspective — it 'lives longer' in the Earth's frame due to time dilation, allowing it to reach the surface. Time dilation is well verified experimentally (GPS clocks, particle accelerators, muon experiments).
• Mass-energy equivalence: E = mc² = γm₀c². Rest energy: E₀ = m₀c² (energy equivalent of rest mass). Kinetic energy (relativistic): K = (γ−1)m₀c² (NOT ½m₀v² except at low speeds). Total energy E = rest energy + kinetic energy = m₀c² + K = γm₀c². Note: at low speeds γ ≈ 1 + v²/(2c²), so K ≈ m₀c²(1 + v²/2c² − 1) = m₀v²/2 = ½m₀v² ✓ (classical KE recovered).

All relativistic formulas reduce to their classical counterparts when v << c. This is called the classical limit or correspondence principle: new physics (relativity, quantum mechanics) must replicate old physics (Newton's laws) in the appropriate limit. For v << c: v²/c² ≈ 0, so √(1−v²/c²) ≈ 1 − v²/(2c²) ≈ 1 (first order). Therefore: (1) m ≈ m₀ (mass is constant — Newton's assumption) (2) p ≈ m₀v (linear momentum) (3) F = dp/dt ≈ m₀(dv/dt) = m₀a (Newton's second law). (4) K ≈ ½m₀v² (classical KE). The entire framework of classical mechanics is an approximation valid when v << c.

• Why we don't notice relativistic effects in daily life: the fastest human-made object is the Voyager 1 spacecraft at ~17 km/s = 1.7×10⁴ m/s. v/c ≈ 5.7×10⁻⁵. γ = 1/√(1−(5.7×10⁻⁵)²) ≈ 1 + (1.6×10⁻⁹) ≈ 1.0000000016. The relativistic correction is less than 2 parts per billion — completely unmeasurable in classical experiments. Even at 10,000 km/s (Earth-Sun distance in about 15 seconds), v/c ≈ 0.033 and γ ≈ 1.0006. Only at v > 0.1c (10% of light speed) do relativistic effects become noticeable (γ ≈ 1.005).
• NEET context: the reason this topic appears at the end of Newton's Laws chapters is to acknowledge the scope of classical mechanics, not to require deep relativistic calculations. NEET tests: (a) the formula m = m₀/√(1−v²/c²) — usually a substitution; (b) the qualitative fact that mass increases with velocity (not decreases); (c) F = dp/dt is the more general form (not F = ma); (d) the concept of rest mass vs observed mass. Complex tensor-form relativistic mechanics is NOT in NEET scope.
• Mass-energy equivalence E = mc² and nuclear physics: although E = mc² is derived from Special Relativity, NEET covers it primarily in the Nuclear Physics chapter (Binding Energy, Mass Defect). When Δm of mass is converted: ΔE = Δm×c². For NEET's Newton's Laws chapter, the focus is on relativistic mass increasing with velocity. The nuclear applications are tested separately.