Kinetic Energy Calculator

**Kinetic energy (non-relativistic):** KE = ½ × m × v² Where: - KE = kinetic energy (J) - m = mass (kg) - v = velocity (m/s) **Worked example: car at highway speed** A 1,500 kg car at 30 m/s (~67 mph). KE = 0.5 × 1500 × 900 = 675,000 J = 675 kJ Equivalent to 0.16 kWh, or the explosive energy of ~160 g TNT. **Velocity dependence (quadratic):** | v (m/s) | KE per kg | |---|---| | 1 | 0.5 J | | 2 | 2 J | | 5 | 12.5 J | | 10 | 50 J | | 20 | 200 J | | 50 | 1,250 J | | 100 | 5,000 J | Doubling v → 4× KE. Tripling v → 9× KE. **Common kinetic energies:** | Object/Situation | KE | |---|---| | Walking human (~1 m/s) | ~40 J | | Running human (~5 m/s) | ~1 kJ | | Cyclist (15 m/s) | ~7 kJ | | Family car (15 m/s) | ~170 kJ | | Family car (30 m/s) | ~675 kJ | | 9mm bullet (350 m/s) | ~500 J | | Rifle bullet (900 m/s) | ~3,500 J | | Train (40 m/s) | ~10 GJ | | Boeing 747 (250 m/s) | ~12 GJ | | ISS in orbit (7.66 km/s) | ~12 TJ | | Earth orbiting Sun (30 km/s) | 2.7 × 10³³ J | **Connection to work:** Work-energy theorem: net work done on object = change in KE. W = ΔKE = KE_final − KE_initial = ½m(v_f² − v_i²) A car braking from 30 m/s to 0: W_brakes = 0 − 675,000 = −675,000 J The brakes do −675 kJ of work — kinetic energy converts to heat in the brake discs. **Energy conservation:** For a falling object (no air resistance): mgh = ½mv² v = √(2gh) For a swinging pendulum: KE_max at bottom = PE_max at top ½mv² = mgh **Rotational kinetic energy:** KE_rot = ½ × I × ω² Where I = moment of inertia, ω = angular velocity. For a rolling object (combined translation + rotation): KE_total = ½mv² + ½Iω² Solid sphere rolling: KE_total = ½mv² × (1 + 2/5) = 0.7mv². **Stopping distance (constant deceleration):** d = v² / (2a) = v² / (2μg) Where μ = friction coefficient. | v (m/s) | mph | Stopping d (μ=0.7) | |---|---|---| | 10 | 22 | 7.3 m | | 20 | 45 | 29.1 m | | 30 | 67 | 65.5 m | | 40 | 89 | 116.5 m | | 50 | 112 | 182 m | Quadratic scaling — doubling speed quadruples stopping distance. **Air drag and kinetic energy:** Energy lost to drag per meter: F_drag = ½ρv²C_dA. At constant speed, engine power needed = F × v = ½ρv³C_dA. Power scales as v³ — going 1.5× faster requires ~3.4× more power. This is why highway fuel economy drops sharply above ~70 mph. **Relativistic kinetic energy:** At low speeds (v << c), classical formula works. At high speeds: KE = (γ − 1) × m × c² Where γ = 1/√(1 − v²/c²). For v = 0.5c: γ ≈ 1.155 → KE = 0.155mc² (vs 0.125mc² classical — 24% higher). For v = 0.99c: γ ≈ 7.09 → KE = 6.09mc² (vs 0.49mc² classical — 12× higher). Why nothing massive can reach c: KE → ∞ as v → c. **Particle physics units (eV):** 1 eV = 1.602 × 10⁻¹⁹ J. Atomic-scale energies often in eV, keV, MeV, GeV, TeV. - LHC proton: 13 TeV total (mostly KE). - Atomic ionization: ~10 eV. - Nuclear binding: ~MeV per nucleon. **Common safety energies:** - **OSHA hand-tool projectile**: < 1 J safe. - **Eye injury threshold**: ~1 J impact. - **Skin penetration**: ~10-50 J/cm² for typical projectile. - **Stopping a person**: 100-1000 J typical absorbing capacity. - **Lethal trauma**: variable, often hundreds to thousands of J localized. **Hydraulic and pneumatic equivalents:** A water jet at velocity v and flow rate Q: P_KE = ½ρQv² Useful for fire-hose analysis, jet propulsion, hydraulic systems. **Bullet kinetic energy and damage:** | Round | Mass (g) | Velocity (m/s) | KE (J) | |---|---|---|---| | .22 LR | 2.6 | 350 | 159 | | 9mm | 8.0 | 360 | 519 | | .357 Magnum | 10.2 | 460 | 1,079 | | .45 ACP | 14.9 | 285 | 605 | | 5.56mm NATO | 4.0 | 990 | 1,960 | | 7.62mm NATO | 9.3 | 850 | 3,360 | | .50 BMG | 42.0 | 880 | 16,300 | Note: damage is more complex than just KE — depends on penetration, deformation, and tissue interaction.