Escape Velocity & Gravitational Potential Calculator

Gravitational Physics Guide

Escape Velocity

Escape velocity is the minimum speed needed to escape a gravitational field without further propulsion. v_escape = √(2GM/R). From Earth's surface: v = √(2 × 6.67×10⁻¹¹ × 5.97×10²⁴ / 6.37×10⁶) = √(1.25×10⁸) = 11.19 km/s ≈ 40,300 km/h. Note: this assumes no atmosphere. Real spacecraft need more speed to overcome air drag. The escape velocity applies to ballistic trajectories — a rocket with constant thrust does not need to reach 11 km/s instantaneously.

Gravitational Field Strength

g = GM/r². At Earth's surface: g = 9.81 m/s². At altitude h: g_h = GM/(R+h)². Gravitational field strength decreases with the square of distance from the centre. At ISS altitude (400km): g = 6.67×10⁻¹¹ × 5.97×10²⁴ / (6.77×10⁶)² = 8.7 m/s². Astronauts experience 'weightlessness' not because there is no gravity (still ~89% of surface gravity) but because they are in free fall — in circular orbit the spacecraft and astronaut fall together, so no normal force acts on them.

Orbital Mechanics

For circular orbit at altitude h: orbital speed v = √(GM/(R+h)). Period T = 2π(R+h)/v = 2π√((R+h)³/(GM)). ISS (400km altitude): v = √(6.67×10⁻¹¹ × 5.97×10²⁴ / 6.77×10⁶) = 7.67 km/s. T = 2π × 6.77×10⁶ / 7670 = 5,541 s = 92.4 minutes. Geostationary orbit: T = 24 hours. r = ∛(GMT²/(4π²)) = 42,164 km from centre = 35,786 km above surface. Kepler's third law: T² ∝ r³ follows directly from the orbital mechanics equations.

Black Holes and Schwarzschild Radius

If a mass is compressed within its Schwarzschild radius, it becomes a black hole. r_s = 2GM/c². Earth: r_s = 2 × 6.67×10⁻¹¹ × 5.97×10²⁴ / (3×10⁸)² = 8.87mm. The Earth would need to be compressed to less than 9mm radius to become a black hole. Sun: r_s ≈ 3km. Stellar black holes: typically 10-30 km Schwarzschild radius, 3-20 solar masses. Supermassive black holes: millions to billions of solar masses. M87* (first black hole photographed): 6.5 billion solar masses, r_s ≈ 19 billion km. The escape