“To infinity and beyond”, may be the catchphrase of Buzz Lightyear (one of the Toy Story heros) but why would someone, or something, need to get to infinity in the first place? Well, one reason could be in order to completely escape the clutches of a gravitational field, and that means travelling at escape velocity.

Gravitational fields extend to infinity, so for Buzz to truly escape one, he really would need to travel at least that distance! Obviously nothing can actually reach infinity, let alone “beyond”, however there's nothing wrong with having a theoretical model which allows us to derive a formula from which we can calculate the required velocity for a projectile (Buzz Lightyear’s space capsule, for example), to travel to infinity from the vicinity of the Earth. 

At less than escape velocity, our projectile will either fall back to Earth (like a stone thrown in the air), or, with enough kinetic energy and a suitable trajectory, go into a closed orbit around the planet. However, at escape velocity, the projectile will, assuming no forces other than gravity are in play, continue on its way until its kinetic energy has been entirely spent.

Whilst you may wish to imagine an Einsteinian gravitational field, a potential well in the fabric of space-time out of which our projectile must climb; in order to find a mathematical formula with which to calculate the required escape velocity for Buzz’s capsule, for our purposes Newton's familiar model of gravity will suffice.

Before proceeding it's important to remember that the concept of escape velocity only applies to intrinsically inert bodies, like cannon balls for example. Rockets, on the other hand, are powered by fuel and are continuously propelled under their own internally generated energy. It only makes sense to talk about escape velocity where the single force involved is that of gravity. A rocket may of course be propelled by its “engine” to a height above a planetary surface where it becomes a projectile (for example, by cutting the fuel supply once beyond the effects of atmospheric drag), having achieved the required escape velocity for that particular altitude, and beyond which fuel is no longer required. 

Actually, even the word velocity is in this regard somewhat of a misnomer, since gravity is a conservative force, and therefore the phrase, escape “speed”, perfectly describes the sought after value. The work done in order for a projectile to overcome a gravitational field, depends simply on its starting and finishing distance from the source of the field and consequently can take any route it wants!

To get our projectile to infinity, it needs to convert kinetic energy to gravitational potential energy. This is basically the opposite of nudging an object (at infinity), into a gravitational field so that it converts gravitational potential energy to kinetic energy, as it falls towards the source. Since the only force involved is gravity, we can invoke the law of conservation of total mechanical energy, which means in this case that the sum of the kinetic energy (\({1 \over 2} mv^2 \)) and gravitational potential energy of the projectile is constant.

In order to find an expression for the projectile’s gravitational potential energy, we could simply use the idea of work (negative in this case, due to the opposing force direction) being equal to force times distance. However, since the Newtonian gravitational force is a function of distance, it must be integrated over the distance required to get the projectile from the Earth to infinity; which, with some simple calculus gets us to \(GMm({1\over r_2}-{1\over r_1})\), where

• \(G\) is the gravitational constant,
• \(M\) is the mass of the gravitational field source (the Earth in our example),
• \(m\) is the mass of Buzz’s capsule,
• and \(r_2\) and \(r_1\) are the destination and starting distances (respectively) from the field source. 

Since \(r_2\) is at infinity, the above expression simply reduces to \(-G{Mm\over r}\), where \(r\) is now the distance from the centre of the gravitational field. 

Don't be worried about the negative sign. It is there because, at infinity, the projectile has zero gravitational potential energy:

\(-G{Mm\over \infty}=0\)

Therefore, at any point between \(r = 0\), and \(r = \infty\), the projectile will have a gravitational potential energy of less than zero: increasingly negative towards, and decreasingly negative away from, the source of the field.

Now we can use the aforementioned conservation law, noting that, since the projectile, at infinity, will have used up all of its kinetic energy and will have zero gravitational potential energy, we can say that at every point in its journey:

\({1 \over 2} mv^2 -G{Mm\over r}=0\)

Therefore,

\({1 \over 2} mv^2 =G{Mm\over r}\)

from which, we can get

\(v = \sqrt {2GM \over r}\)

the required formula for escape speed, which is a function of both the mass of the source of the gravitational field and the projectile’s distance from it. Please note, by the way, the complete lack of dependence on the mass of the projectile.

Obviously, whatever the scenario, whether a comet in an open hyperbolic orbit around a star, or the Voyager space probes zipping away from the solar system, there are as many infinite gravitational fields in the Universe as there are celestial bodies, so there's really no escaping gravity. As a final point of interest, it's worth noting that with a suitably large mass and a commensurately small radius, the escape speed can be greater, even, than the speed of light, in which scenario we have a black hole on our hands - Buzz beware!

Images

• Buzz Lightyear (Toy Story character) by http://www.moviestillsdb.com/movies/toy-story-3-i435761/20c11d67, Fair use, https://en.wikipedia.org/w/index.php?curid=44184427