# Escape Velocity Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Feb 13, 2023

The escape velocity calculator is a tool that you can use to find what speed an object needs to gain in order to leave the surface of any celestial body, opposing its gravity. This article will explain in detail how to calculate escape velocity and the first cosmic velocity. It will also provide you with a thorough explanation of the escape velocity equation.

## Escape velocity equation

The escape velocity formula is independent of the properties of the escaping object. The only thing that matters is the mass and radius of the celestial body in question:

$\small V_e = \sqrt{\frac{2 \mathrm G M}{R}}$

$M$ is the mass of the planet, $R$ is its radius, and $\mathrm G$ is the gravitational constant. It is equal to $\small\mathrm G = 6.674 × 10^{-11}\ \mathrm{N·m²/kg²}$.

The formula for escape velocity, also known as the second cosmic velocity, is derived directly from the law of conservation of energy. At the moment of launch, the object has some potential energy $PE$ and some kinetic energy $KE$. The energy at launch $LE$ can be hence presented as follows:

$\small PE + KE = -\frac{\mathrm G M m}{R} + \frac{1}{2}mv^2$

where $m$ is the mass of the starting object, and $v$ is the escape velocity.

When the object finally escapes, it is located so far from the planet that its potential energy is equal to zero. Also, it can have virtually no speed, so its kinetic energy is also equal to zero. That means that the total final energy is equal to:

$\small PE + KE = 0 + 0 = 0$

Because the total energy must be conserved, it means that the initial energy is also equal to zero. Simplifying the first equation, we get:

\small \begin{align*} 0 & = - \frac{\mathrm G M m}{R} + \frac{1}{2}mv^2 \\ v & = \sqrt{\frac{2 \mathrm G M}{R}} \end{align*}

Check our kinetic energy calculator and potential energy calculator for more details on the topic of energy.

## How to calculate escape velocity

Just follow these steps, and you will have it calculated in no time!

1. Determine the mass of the planet. For example, the mass of Earth is equal to $5.9723 \times 10^{24} \ \mathrm{kg}$.
2. Determine the radius of the planet. For instance, the radius of Earth is $6,371\ \mathrm{km}$.
3. Substitute these values in the escape velocity equation $v = √(2\mathrm GM/R)$.
4. Calculate the result. It the case of Earth, the escape velocity is equal to $11.2 \ \mathrm{km/s}$.
5. Check whether the result is correct using out escape velocity calculator.

## First cosmic velocity

You probably noticed that this calculator gives you an additional value – the first cosmic velocity. What is it, and what is the difference between this value and the escape velocity?

The first cosmic velocity is the velocity that an object needs to orbit the celestial body. For example, all satellites need to have this velocity in order not to fall back to the surface of Earth. It is equal to the escape velocity divided by the square root of 2. The full formula looks like this:

$\text{first cosmic velocity} = √(M\mathrm G/R)$

You already know what is the second cosmic velocity, also known as the escape velocity – the speed required to leave the surface of a planet. For instance, this is the velocity of space rockets.

## Typical values

You can find the escape velocities of all planets of the Solar System (and of the Moon) below. Perhaps you can use the escape velocity equation 'backward' to calculate their masses and radii? Try to find the first cosmic velocities of these planets, too!

• Mercury: 4.3 km/s
• Venus: 10.3 km/s
• Earth: 11.2 km/s
• Moon: 2.4 km/s
• Mars: 5.0 km/s
• Jupiter: 59.6 km/s
• Saturn: 35.6 km/s
• Uranus: 21.3 km/s
• Neptune: 23.8 km/s

Make sure to check out our velocity calculator!

Bogna Szyk
Mass (M)
Earths
R⊕
First cosmic velocity
mi/s
Escape velocity (vₑ)
mi/s
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