Escape Velocity Calculator: v = sqrt(2GM / r)

The speed needed to leave a planet's gravity without any more push.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • v_e = \sqrt{\frac{2GM}{r}}
  • M = \frac{v_e^2r}{2G}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the escape velocity: escape velocity = (sqrt(2G) x mass of the planet or moon^0.5) / (radius (distance from the centre)^0.5)
  2. Put the values in (every value in its SI unit): escape velocity = (sqrt(2G) x 5.972e+24^0.5) / (6.371e+6^0.5)
  3. The answer: escape velocity: 11186 m/s
  4. The same answer in other units: 11.186 km/s; 40269.5 km/h; 25022.3 mph

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Radius (distance from the centre); Escape velocity (leave it as it is if this is what you are finding) 11186 m/s; Mass of the planet or moon (leave it as it is if this is what you are finding) 5.972E+24 kg.

  1. The formula, arranged to find the radius (distance from the centre): radius (distance from the centre) = ((sqrt(2G) x mass of the planet or moon^0.5) / (escape velocity))^2
  2. Put the values in (every value in its SI unit): radius (distance from the centre) = ((sqrt(2G) x 5.972e+24^0.5) / (11186))^2
  3. The answer: radius (distance from the centre): 6.37097e+6 m
  4. The same answer in other units: 6370.97 km; 3958.74 miles

Common questions

What is the escape velocity of the Earth?

About 11.19 km/s, or 40 300 km/h, from the surface.

Does the mass of a rocket change the escape velocity?

No. A heavier rocket needs more fuel to reach it, but the speed needed is the same.

What is the escape velocity of the Moon?

About 2.38 km/s.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics text (gravitation). G from the 2018 CODATA values.

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