Escape Velocity Calculator
Calculate the ideal speed needed to escape a spherical body's gravity from its surface, plus the corresponding first cosmic (circular orbital) velocity.
Inputs
Required. Positive kilograms; scientific notation such as 5.9723e24 is accepted.
Required. Positive radius from the body's center to the launch point, in kilometers.
Live results
Uses G = 6.674 × 10⁻¹¹ m³·kg⁻¹·s⁻² to match the calculation convention documented for this model.
How to use the Escape Velocity Calculator
What this calculator does
This calculator estimates two idealized speeds for a spherical gravitating body. Escape velocity (vₑ) is the minimum initial speed a non-propelled object would need at the stated radius to reach arbitrarily far away with zero remaining speed in the ideal two-body model. First cosmic velocity is the circular orbital speed at that same radius. The calculation uses only the body's mass, radius, and Newton's gravitational constant; it does not model atmospheric drag, thrust profiles, planetary rotation, other bodies, terrain, relativity, or practical launch trajectories. NASA's explanation of escape velocity and its dependence on mass and distance from the center is a useful conceptual companion.
When to use it
Use the calculator to check a planet, moon, asteroid, or hypothetical spherical body's ideal surface escape speed; compare how a denser or more compact body changes the required speed; verify astronomy or introductory mechanics exercises; or estimate the circular orbital speed immediately above a body's surface under the same simplified assumptions.
How to calculate
- The calculator opens with an Earth-like demonstration: mass 5.9723e24 kg and radius 6,371 km. Results and a validated example XLSX are available immediately.
- Replace Mass (M) with the body's mass in kilograms. Scientific notation is accepted, so a value such as 7.342e22 is valid.
- Replace Radius (R) with the distance from the body's center to the launch point in kilometers. For a surface calculation, use the relevant physical radius.
- Read Escape velocity (vₑ) and First cosmic velocity, both shown in km/s. The values update as you type when both inputs are valid.
- Select Download Excel to export the current inputs, outputs, formula, units, and model note to a real XLSX workbook. Reset clears the demonstration data and results; Download Excel is then disabled until a complete valid mass-and-radius pair is entered again.
Input guide
Mass (M) is required and must be a finite number greater than zero, expressed in kilograms. The field accepts ordinary decimal notation, grouped thousands using commas in standard English formatting, and scientific notation such as 5.9723e24. Increasing mass while holding radius fixed raises both velocities in proportion to the square root of mass. A common mistake is entering a mass in grams or metric tons without converting to kilograms, which changes the result by a large factor.
Radius (R) is required and must be a finite positive number in kilometers. The startup example uses 6,371 km. The model converts this value to meters internally before applying the formula. Increasing radius while keeping mass fixed lowers both velocities according to the inverse square root of radius. Do not enter diameter in place of radius, and do not mix meters with kilometers. For an altitude calculation, the relevant R is the body's physical radius plus altitude measured from the center – not altitude alone.
Output guide
Escape velocity (vₑ) is the primary estimate in kilometers per second. It rises with mass and falls with radius. A very low value describes weak gravity at the chosen radius; a higher value indicates that a greater initial speed is needed to escape in the idealized model. First cosmic velocity is the circular orbital speed at the same radius, also in km/s. It is exactly the escape velocity divided by √2 in this Newtonian model. The header pills mirror the current mass, radius, and escape speed so you can scan the active scenario quickly. These are mathematical estimates, not launch-vehicle performance recommendations.
Worked example
With the startup values M = 5.9723 × 10²⁴ kg and R = 6,371 km = 6.371 × 10⁶ m, the model evaluates vₑ = √(2 × 6.674 × 10⁻¹¹ × 5.9723 × 10²⁴ / 6.371 × 10⁶). The result is about 11.186 km/s. Dividing by √2 gives the first cosmic velocity, about 7.910 km/s. These values are consistent with the familiar approximately 11.2 km/s terrestrial escape-speed figure. For comparison data across planets, see JPL's planetary physical parameters and escape velocities.
Learn more
The gravitational constant is a measured physical constant. The current CODATA recommendation can be checked at the NIST reference for the Newtonian constant of gravitation. This calculator intentionally uses 6.674 × 10⁻¹¹ m³·kg⁻¹·s⁻² because that is the stated convention for the model reproduced here; the tiny difference from the latest CODATA value is much smaller than the displayed three-decimal km/s precision for the Earth example.
How the model works
The escape-speed equation follows from conservation of mechanical energy. At the launch radius, gravitational potential energy is negative and kinetic energy is positive. The threshold escape case sets the total energy to zero: the object arrives infinitely far away with vanishingly small residual speed. Solving that energy balance for launch speed gives vₑ = √(2GM/R). Because the mass of the escaping object appears in both the kinetic- and potential-energy terms, it cancels; in the ideal model, a small probe and a large spacecraft have the same escape speed from the same location.
The first cosmic velocity uses v₁ = √(GM/R). That is the speed required for a circular orbit at radius R when the central body is spherical and other perturbations are ignored. The two equations differ only by the factor √2, so their ratio never changes as long as the same mass and radius feed both results.
Interpretation and limitations
Real launches are more complicated. A rocket does not normally receive one instantaneous velocity impulse at the surface. It burns fuel over time, loses energy to atmospheric drag and gravity during ascent, may benefit from planetary rotation, and targets an orbit or interplanetary trajectory rather than a mathematical escape-to-infinity boundary. The calculator is therefore most useful as a physics identity and scale estimate. It also assumes Newtonian gravity. For extremely compact objects, relativity becomes essential; NASA discusses the connection between escape-speed intuition and the Schwarzschild-radius idea for black holes.