What is the escape velocity formula?

Escape velocity is v = √(2GM/r), where G is the gravitational constant 6.674e-11 N·m²/kg², M is the body mass in kilograms and r is the radius in metres. It is the slowest speed an object needs to leave the body's gravity without any further push. For Earth, plugging in M = 5.972e24 and r = 6.371e6 gives about 11 186 m/s, or 11.2 km/s. The formula assumes no air drag and that you start from the surface.

Why is Earth's escape velocity 11.2 km/s?

Earth holds a mass of 5.972e24 kg inside a radius of 6.371e6 m. Feeding those into v = √(2GM/r) gives √(2 × 6.674e-11 × 5.972e24 / 6.371e6) ≈ 11 186 m/s, which is 11.2 km/s or about 25 000 mph. That is the speed a rocket must reach at the surface to coast away forever. Real rockets carry extra speed to fight air drag and gravity losses during the climb, so they are not magically slow at launch.

How does escape velocity relate to orbital velocity?

Orbital velocity (the first cosmic velocity) is √(GM/r), the speed for a circular orbit skimming the surface. Escape velocity √(2GM/r) is exactly √2 ≈ 1.414 times larger. So Earth's surface orbital speed is about 7.9 km/s and the escape speed is 11.2 km/s. The neat takeaway: if a satellite is circling, you only need to boost it by a factor of 1.414 to send it out of the system entirely.

Is a black hole's escape velocity really the speed of light?

At the event horizon, yes. A black hole packs so much mass M into so small a radius r that √(2GM/r) reaches c, the speed of light, about 3e8 m/s. That radius is the Schwarzschild radius, 2GM/c². Inside it the required escape speed exceeds light, which nothing can reach, so not even light escapes. You can see the trend in this tool: shrink the radius or pile on mass and the velocity climbs fast.

What raises escape velocity, more mass or a smaller radius?

Both, but in different ways. Escape velocity grows as the square root of mass, so quadrupling M only doubles v. It also grows as one over the square root of radius, so squeezing the same mass into a quarter of the radius doubles v. That is why the Sun, despite its enormous radius, still beats every planet at about 617 km/s — its mass is overwhelming. Try editing the mass and radius fields here to watch each effect on its own.

Does escape velocity depend on the object being launched?

No. The mass of the rocket, ball or atom cancels out of the energy equation, so escape velocity depends only on the body you are leaving, never on what is leaving. A pebble and a spaceship both need 11.2 km/s to escape Earth from the surface. What differs is the energy and fuel required to reach that speed, since heavier objects need more push to accelerate. The number this tool reports is purely a property of the planet, moon or star.

Verify a launch budget for a model-rocket or Kerbal-style mission plan

Punch in a moon's mass and radius to read its escape speed, then check whether your stage's delta-v has any margin to leave it. Seeing 2.38 km/s for the Moon versus 5.03 km/s for Mars tells you instantly which body your design can actually break free of.

Settle the radius-versus-mass argument with live numbers

Keep one field fixed and edit the other. Quadruple the mass and the escape speed only doubles; shrink the radius to a quarter and it doubles too. Watching the root-2 step-by-step output update makes the square-root scaling click far faster than re-deriving it.

Escape Velocity Calculator — v = √(2GM/r)

v = √(2GM/r) — escape + orbital velocity for any planet, moon or star, in m/s, km/s and mph — browser-only

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Body mass M

Radius r

Body presetSpeed unitG = 6.674e-11 N·m²/kg²

Escape velocity11.186 km/s

Orbital velocity (first cosmic)7.9095 km/s

v_escape = √2 × v_orbit ≈ 1.414 × v_orbit

Formula steps

v = √(2·G·M / r)
v = √((2 × 6.674e-11 × 5.9720e+24) / 6,371,000)
v = √(1.2512e+8)
v_escape = 11,186 m/s
v_orbit = 7,909.5 m/s = v_escape / √2

What this tool does

Free escape velocity calculator that turns a body mass and radius into the speed needed to leave its gravity for good. The formula is v = √(2GM/r), where G is the gravitational constant 6.674e-11, M the mass in kilograms and r the radius in metres. Type Earth (5.972e24 kg, 6.371e6 m) and you get the textbook 11.2 km/s; pick the Moon and it drops to about 2.38 km/s because the Moon holds less mass over a smaller radius. The tool also reports the orbital velocity, the first cosmic velocity √(GM/r) that keeps a satellite circling at the surface, and shows on screen that escape velocity is always exactly √2 times larger. One-click presets cover Earth, the Moon, Mars, the Sun and Jupiter, results switch between metres per second, kilometres per second and miles per hour, and every step of the square-root math is laid out so you can check the work. Everything runs in your browser with a shareable link that reopens the exact body. 100% client-side, no upload.

Tool details

Input: Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Escape Velocity Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Verify a launch budget for a model-rocket or Kerbal-style mission plan
Punch in a moon's mass and radius to read its escape speed, then check whether your stage's delta-v has any margin to leave it. Seeing 2.38 km/s for the Moon versus 5.03 km/s for Mars tells you instantly which body your design can actually break free of.
Settle the radius-versus-mass argument with live numbers
Keep one field fixed and edit the other. Quadruple the mass and the escape speed only doubles; shrink the radius to a quarter and it doubles too. Watching the root-2 step-by-step output update makes the square-root scaling click far faster than re-deriving it.

Common pitfalls

Treating escape velocity as a direction or a launch angle. It is a speed only; reach 11.2 km/s pointed any way above the horizon and you escape Earth.
Forgetting it is measured from the surface. Start higher up where r is larger and the required escape speed is lower than the surface figure the tool reports.
Reading the result as the speed needed to reach orbit. Orbiting takes the smaller circular-orbit value; escaping for good takes the root-2-larger number shown beside it.

Privacy

The mass, radius, unit choice and every step of the root-2 GM/r math are computed by JavaScript inside your own browser tab, with no server call and nothing written to any log. The only data that travels is the shareable link, which carries your inputs in its query string; skip sharing it if you would rather keep even those numbers to yourself.

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Escape Velocity Calculator — v = √(2GM/r)

Formula steps

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Escape Velocity Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Verify a launch budget for a model-rocket or Kerbal-style mission plan

Settle the radius-versus-mass argument with live numbers

Common pitfalls

Privacy

FAQ

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