The first problem facing a potential trip to Mars is leaving Earth. Specifically, this problems deals with the enormous amount of energy necessary to break free from the Earth’s gravitational field and start traveling towards Mars, or anywhere else in the Solar System. To find out what energy, and therefore speed, is necessary to escape Earth’s gravity, let us consider the energy of a rocket at Earth’s surface:

Energy is the sum of kinetic and potential energies. Here, vinitial is the initial velocity, mrocket is the mass of the rocket, and Mearth and Rearth are the mass of the Earth and the radius of the Earth. Now, because the energy of the rocket is constant as it travels upward, we can equate the energy of the rocket at the surface to the energy of the rocket at its maximum altitude:

Here, vfinal is the final velocity and rmaximum is the maximum height. However, at its maximum height, vfinal = 0, so the equation becomes

Solving for vinitial, we have

Setting rmaximum = infinity, which is the condition for gravitational escape, vinitial becomes vescape and we have

The same logic can be applied to any planet, so the equation for escape velocity can be generalized to

Thus, the escape velocity from any planet depends on the mass of the planet and the radius of the planet.

For example, let us assume that we have a spacecraft on Earth that we are trying to send into space. Mearth = 5.98x1024 kg, and Rearth = 6.37x106 m, so we get:

TNow, let us assume astronauts have successfully completed their mission on Mars and need to calculate the escape velocity on Mars so they can travel back to Earth. Mmars = 6.42x1023 kg, and Rmars = 3.397x106 m, so we get: