Life gets a bit stressful at times and all you wanna do is… escape. So you buy the cheapest retail space rocket you can from NASA and blast off into the universe at your local launch pad, leaving the Earth behind in your wake, a pale blue dot on the everlasting black canvas of spacetime. You leave behind all your worries, all your problems… your belongings, your family, your friends. Stretching your arms out, you lie down to relax and enjoy a weekly release of The Nexus, your favourite science blog. But as you read, a sudden epiphany strikes your mind. Falling down to your knees, you shake your fists at the emptiness outside. You realise that… you can never truly escape.
I’m not quite sure what happened right there. Please do ignore my feeble attempt at artistic writing. I should honestly just stick to science. Good ol’ science. That short prologue may seem a little frightening (if you were wanting to escape the Earth for any reason) and it tends to be that frightening things turn out not to be real. However, I can say ,with science as my witness, that you cannot actually leave the Earth. In fact, you can’t leave anything, not a single thing in the universe. This is because gravity, that well known phenomenon which pulls us all together, acts over an infinite distance. No matter how large a distance you bridge between two massive objects, they will always have a gravitational attraction towards each other. Theoretically speaking, you would only be able to surpass gravity by reaching an infinite distance away from the object, which is, of course, impossible. Or is it? Yeah, but not in maths.
Using mathematical assumptions and modelling, we are able to therefore calculate the escape velocity for a projectile which is trying to escape the gravitational field of a given body, like the Earth. If you imagine an object resting just at the edge of the Earth’s infinitely large ‘sphere of gravitational influence’ and you let it drop down towards the center of Earth, the gravitational attraction will increase as it gets closer and closer to Earth, so it will accelerate faster and faster, eventually striking the surface of the Earth with some velocity. Now, if we reverse this process and imagine firing the same object from the surface of the Earth upwards with that same velocity the falling object had at the instant of impact, the object will slow down as it rises, being pulled backwards by gravitational forces. The projectile will have just enough velocity to reach infinity, stopping when it arrives there, and the kinetic energy will equal zero. This velocity we fired the object at is precisely the velocity required to escape the gravitational influence of the Earth, no more and no less.
For those of you more mathematically inclined, I will include below a formula for finding the escape velocity of a projectile. It can easily be obtained by considering the total energy at infinity, and substituting formulas for both gravitational potential and kinetic energy.
G is the universal gravitational constant in N m2 kg-2, M is the mass of the body to be escaped in kg, and r is the radius of said body in m.
This idea of having to reach infinity in order to fully escape gravity is indeed a very difficult one to grasp. That’s part of the wonder of studying science. Even though nothing really makes any sense, it makes other things make a bit more sense, and that’s always handy.