Editing Escape velocity (section)

=== Conservation of energy ===
[[Image:RIAN archive 510848 Interplanetary station Luna 1 - blacked.jpg|thumb|[[Luna 1]], launched in 1959, was the first artificial object to attain escape velocity from Earth.<ref>{{Cite web |url=https://nssdc.gsfc.nasa.gov/nmc/spacecraft/display.action?id=1959-012A |title=NASA – NSSDC – Spacecraft – Details<!-- Bot generated title --> |access-date=21 August 2019 |archive-date=2 June 2019 |archive-url=https://web.archive.org/web/20190602031816/https://nssdc.gsfc.nasa.gov/nmc/spacecraft/display.action?id=1959-012A |url-status=live }}</ref> (See [[List of Solar System probes]] for more.)]]

The existence of escape velocity can be thought of as a consequence of [[conservation of energy]] and an energy field of finite depth. For an object with a given total energy, which is moving subject to [[conservative force]]s (such as a static gravity field) it is only possible for the object to reach combinations of locations and speeds which have that total energy; places which have a higher potential energy than this cannot be reached at all. Adding speed (kinetic energy) to an object expands the region of locations it can reach, until, with enough energy, everywhere to infinity becomes accessible.

The formula for escape velocity can be derived from the principle of conservation of energy. For the sake of simplicity, unless stated otherwise, we assume that an object will escape the gravitational field of a uniform spherical planet by moving away from it and that the only significant force acting on the moving object is the planet's gravity. Imagine that a spaceship of mass ''m'' is initially at a distance ''r'' from the center of mass of the planet, whose mass is ''M'', and its initial speed is equal to its escape velocity, ''v''{{sub|e}}. At its final state, it will be an infinite distance away from the planet, and its speed will be negligibly small. [[Kinetic energy]] ''K'' and [[gravitational potential energy]] ''U''<sub>g</sub> are the only types of energy that we will deal with (we will ignore the drag of the atmosphere), so by the conservation of energy,
: <math>(K + U_\text{g})_\text{initial} = (K + U_\text{g})_\text{final}</math>

We can set ''K''<sub>final</sub> = 0 because final velocity is arbitrarily small, and {{nowrap|''U''<sub>g</sub>{{hsp}}<sub>final</sub>}} = 0 because final gravitational potential energy is defined to be zero a long distance away from a planet, so
: <math>\begin{align}
  \Rightarrow {} &\frac{1}{2}m{v_\text{e}}^2 + \frac{-GMm}{r} = 0 + 0 \\[3pt]
  \Rightarrow {} &v_\text{e} = \sqrt{\frac{2GM}{r}}
\end{align}</math>