Escape Velocity Calculator
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Everything in space is a delicate balance. A moving object, like a rocket or an asteroid, has inertia and wants to travel in a straight line forever. However, the immense gravitational force of a massive body (like a planet or a star) constantly attracts it, trying to make it fall.
From the interaction between these two forces arise the fundamental concepts of orbital motion.
What is Orbital Velocity?
Orbital velocity (vorb) is the perfect "equilibrium point"—the exact speed at which an object must move laterally so that the curvature of its fall (caused by gravity) perfectly matches the curvature of the planet.
Imagine you're on an incredibly tall tower and you fire a cannonball.
If you fire slowly, it falls to Earth.
If you fire faster, it travels farther before falling.
If you fire at the exact orbital velocity, the cannonball "falls" perpetually, but the Earth curves "away" from it at the same rate. The cannonball never lands: it's now in orbit.
It's in a constant state of free fall, but falling around the planet, not towards it.
What is Escape Velocity?
Now, what happens if you fire that cannonball even faster?
Escape velocity (ve) is the point of no return; it is the minimum speed at which the object's kinetic energy is so great that the pull of gravity, although slowing it down, is never enough to stop it or change its path back.
The object breaks its closed orbit and escapes the planet's gravitational "well," traveling indefinitely into deep space on an open trajectory (parabolic or hyperbolic).
The Absolute Limit: Schwarzschild Radius
Physics takes this concept to its most fascinating extreme. How dense would an object have to be for its escape velocity to equal the speed of light (the theoretical maximum speed in the universe)?
The Schwarzschild radius (Rs) answers this question. It is the radius to which you would have to compress the entire mass of a body for its surface gravity to be so intense that not even light could escape. This is the threshold that defines the event horizon of a black hole.
These three metrics are not just numbers; they are the principles that dictate how satellites orbit our world, how we send probes to other planets, and how we understand the most extreme objects in the cosmos.
Formulas used by the calculator
Orbital velocity: vorb = √( G·M / (R + h) )
Escape velocity: ve = √( 2·G·M / (R + h) ) = √2 · vorb
Schwarzschild radius: Rs = 2·G·M / c²
Where M is the mass of the body in kg, R its radius in m, h the altitude above the surface in m, G = 6.67430 × 10⁻¹¹ m³/(kg·s²) is the universal gravitational constant and c = 299,792,458 m/s is the speed of light. With h = 0 you get the values at the surface.
Worked example: the International Space Station (ISS)
Data: M = 5.972 × 10²⁴ kg (Earth), R = 6.371 × 10⁶ m, h = 408 km = 4.08 × 10⁵ m.
1. Distance to the center: r = R + h = 6.371 × 10⁶ + 4.08 × 10⁵ = 6.779 × 10⁶ m.
2. Product G·M = 6.67430 × 10⁻¹¹ × 5.972 × 10²⁴ = 3.986 × 10¹⁴ m³/s².
3. Orbital velocity: vorb = √(3.986 × 10¹⁴ / 6.779 × 10⁶) = 7,668 m/s = 7.668 km/s (about 27,600 km/h; the ISS circles the Earth every 92 minutes).
4. Escape velocity at that altitude: ve = √2 × 7.668 = 10.844 km/s.
5. Schwarzschild radius of the Earth: Rs = 2 × 3.986 × 10¹⁴ / (299,792,458)² = 8.87 × 10⁻³ m = 8.87 mm. The whole mass of the Earth would have to fit in a marble-sized sphere to become a black hole.
Escape and orbital velocity at the surface of each body
Moon: ve = 2.38 km/s · vorb = 1.68 km/s
Mercury: ve = 4.25 km/s · vorb = 3.01 km/s
Mars: ve = 5.03 km/s · vorb = 3.55 km/s
Venus: ve = 10.36 km/s · vorb = 7.33 km/s
Earth: ve = 11.19 km/s · vorb = 7.91 km/s
Jupiter: ve = 60.20 km/s · vorb = 42.57 km/s
Sun: ve = 617.77 km/s · vorb = 436.83 km/s
Schwarzschild radius: Earth 8.87 mm, Jupiter 2.82 m, Sun 2.95 km.
Frequently asked questions
Why doesn't escape velocity depend on the mass of the rocket? Because kinetic energy (½·m·v²) and gravitational potential energy (G·M·m / r) are both proportional to the object's mass m; when you equate them, m cancels out. A pebble and a satellite need the same speed to escape.
How are orbital velocity and escape velocity related? At the same distance from the center, escape velocity is always √2 ≈ 1.414 times the circular orbital velocity.
Why do they decrease with altitude? Both depend on 1/√(R + h): the farther away you are, the weaker the pull. In geostationary orbit (h = 35,786 km) the orbital velocity is only 3.07 km/s and the escape velocity 4.35 km/s.
Does a rocket have to reach 11.2 km/s to leave the Earth? Escape velocity is the speed needed by an object that receives no further thrust, ignoring air resistance. A rocket with its engines running could climb more slowly, but it would burn far more fuel; that is why missions gain that energy within a few minutes and above the dense atmosphere.
What does the Schwarzschild radius represent? It is the radius of the event horizon: if all the mass of a body were squeezed inside that sphere, not even light could escape and the object would be a black hole.
The formula for escape velocity is:
