Followup on escape velocity

Mike said:

Seven miles per second. That's the Earth's escape velocity, and the energy required to escape its gravity well...any slower and you'll fall back to Earth.

Karl wrote:

That's always confused me.

Sure, if you get moving to 7 m.p.s, then you'll escape without any further power. But couldn't you be going slower than that and still escape, as long as you continue to provide thrust?

Sure, that's possible, too. In general, though, orbital mechanics equations are generally done with instantaneous velocity changes (i.e. you suddenly go from zero to seven miles per second). This is for two reasons:

1) The orbital equations are just a whole lot simpler to work using instantaneous changes in velocity - it's only algebra. With acceleration over a prolonged period, though, it suddenly becomes a calculus equation.

2) Until very recently, velocity changes to get into and subsequently alter orbits of spacecraft were only done with chemical propellants. Since these are short-lived accelerations, they can be treated as instantaneous to first-order. This is particularly the case for large booster rockets, where once you start the rocket, the whole thing keeps going until it's finished.

Now, reason #2 has recently changed with the creation of spacecraft with ion engines. These engines work on a very different principle...they continuously accelerate individual atoms past an electrified grid, which means very small thrust but over very long periods of time. For more info on this, check out NASA's FAQ on Ion propulsion currently featured on such spacecraft as the Dawn mission to asteroid Vesta.

What you're ultimately trying to do with escape velocity is overcome the Earth's gravity well by raising your potential energy to that of an object at an infinitely far distance. Even though it sounds counter-intuitive, it turns out that this is actually a finite quantity of energy because as distance increases to very large values, the force of gravity becomes infinitesimally small.

As long as you can get your kinetic energy moving away from the Earth equal to this potential energy at infinity, you can escape. Seven miles per second is the oft-quoted figure because that velocity provides a body with enough kinetic energy to be equal to the difference in potential energy between Earth's surface and an object at infinity.

So, to be clear, you ultimately don't need to be going seven miles a second to begin with. You can alter your velocity on the way up however you please...but in the end, you'll have to end up spending at least as much energy as you would've going 7 miles per second initially.