Escape Velocity Calculator

Escape velocity is derived from the principle of conservation of energy, where the kinetic energy of an object equals its gravitational potential energy. The fundamental formula used by this calculator is v = sqrt(2 * G * M / r), where v represents the escape velocity, G is the universal gravitational constant approximately equal to 6.674 x 10^-11 m^3 kg^-1 s^-2, M is the mass of the celestial body, and r is the radius from the center of mass. Additionally, the tool calculates the first cosmic velocity, which is the orbital speed needed for a circular orbit at that altitude, found by dividing the escape velocity by the square root of two.

Let us calculate the escape velocity for a hypothetical planet with a mass of 4.5 x 10^24 kg and a radius of 6,000,000 meters. First, multiply the gravitational constant (6.674 x 10^-11) by the mass (4.5 x 10^24), which yields approximately 300,330. Next, multiply this product by two to get 600,660. Divide this result by the radius (6,000,000 meters) to arrive at 0.10011. Finally, take the square root of 0.10011, resulting in an escape velocity of approximately 11,185 meters per second, or about 11.18 km/s. To find the first cosmic velocity for this planet, divide 11.18 km/s by the square root of two, yielding roughly 7.91 km/s.

When performing orbital mechanics calculations, always convert your measurements into standard SI units: kilograms for mass, meters for radius, and meters per second for velocity. Keep in mind that escape velocity assumes an idealized vacuum with zero atmospheric friction; in reality, atmospheric drag requires rockets to expend extra energy during launch. Additionally, the starting point matters—launching from a high-altitude mountaintop slightly reduces the required radius, marginally lowering the necessary escape velocity.