I understand what escape velocity is, but am wondering WHY it is. (I know…. break the Earth’s gravitational pull and all that.) My question is…. why can’t I (in appropriate space craft of course) just leave the atmosphere at 1 mph? As long as I maintain my 1 mph, shouldn’t I escape the Earth’s gravity (I know, I know… the Earth’s gravity is always pulling on me, no matter how far I am from it)? Just answer the question, and quit your nit picking! Dang… 
Here’s your answer: Yes, you can. Keep going at 1 MPH, and you can get as far away from Earth as you want. But the minute you shut the engine off you will fall back into the gravity well either of Earth or another astronomical body.
Well, then… I suppose I have no idea what escape velocity is. I thought escape velocity was a certain speed (and angle) you had to achieve to leave the planet. I thought that if you did not achieve this speed, you would not make it out of the atmosphere. Is that wrong?
Wouldn’t it take an inordinate amount of power to create a 1mph thrust that would continue at a constant speed though?
Well, unfortunately it is. Escape velocity is the speed necessary to escape the earth’s gravity, and become a independent body. Otherwise, you’re meerly jumping. (albeit a long jump)
You can. Escape velocity only applies if you stop accelerating before the net gravitational pull stops pulling you “back”, and then try to coast the rest of the way.
If you do this, and don’t accelerate to a high enough speed, your speed away from the earth will slow down to zero before the net gravitational pull is no longer pulling you back, and so you start to fall back.
If you do accelerate to a high enough speed (this is what escape velocity is) and then quit accelerating, your velocity away from the earth will still be above zero when the net pull of gravity is no longer toward the earth, and you can then coast onward.
However, if you can continually apply some acceleration even slightly higher than the net gravitational pull toward the earth, you can move as slowly as you like away from the earth for as long and far as you like.
And as long as you have some velocity away from the earth when the net gravitational pull is no longer toward the earth, you can continue on away from the earth at your slow speed with no further force applied.
(Of course the above applies to any pair of bodies, not just to the earth and a hypothetical spacecraft. I just continued using the given example.)
Ugly
I believe the confusion is due to the fact that gravitational pull becomes weaker as you get further from the object. The figure for escape velocity is escape velocity from the Earth’s surface. As you get further from the Earth, escape velocity gets less and less. There probably is a point where the escape velocity is 1 MPH.
However, the energy required to get to that point is considerable – probably the same if not more than starting out at escape velocity from the surface. So there’s no advantage to climbing up and using a lower escape velocity. Might as well put the pedal to the metal and hit it at once.
Neglecting friction, the minimum energy to get to an escape velocity of 1 MPH at a speed of 1 MPH is precisely the same as the energy required to get up to escape velocity at the surface, pretty much by definition.
Again neglecting friction, if you are flying away from a large body at precisely escape velocity than you will continue travel at escape velocity the whole way out. You will be continuously slowed down of course, but the escape velocity will diminish at the same rate. So indeed, if you started at the surface you would eventually reach that 1 MPH point.
Escape velocity is the speed a projectile needs to leave the gravity well from where it is at. This is not the same as just leaving the atmosphere or going into orbit. And that is a projectile not a rocket.
Even though the earths gravity well is technicly infinite, it drops off as with distance(squared) so the net amount of energy you need to have to leave and keep on coasting forever is finite. Thats your esape velocity.
If your in a rocket, you can leave as slowly as you please, but eventually you run out of fuel. When that happens, if you dont have escape velocity from that point (the higher you go, the lower EV is), you will either go into orbit or make nice crater.
This may seem a non sequitor, but this discussion reminds me of the movie “Mouse on the Moon” (the sequel to “The Mouse that Roared”). In MOOM, a scientist from the small country of the Duchy of Grand Fenwick invented a rocket motor that supplied a small but almost infinitely steady thrust. This allowed him to develop a rocket that would v-e-r-y s-l-o-w-l-y travel to the moon. The sight of this homemade rocket lifting off at a top speed of 20 mph was hilarious.
Anyway, I was trying to make a point (I think). That is, a small and steady accelleration is just as effective as a large force in escaping the earth’s pull.
Falcon2 makes the critical distinction here - “escape velocity” refers to a projectile, i.e. a passive hunk of stuff that relies on its momentum to keep moving, not a rocket that provides thrust & therefore accelerates for reasons other than gravity. At less than escape velocity, a projectile will adopt an elliptical orbit (if fired “straight up”, it will come “straight down”). As you increase the projectile’s initial speed, it gets a parabolic orbit, meaning that its velocity at an infinite distance will be zero. (Still moving? Well, you’re not at infinity yet). This is the definition of escape velocity. If you go faster than escape velocity, you’ll be adopting a hyperbolic orbit and your velocity at infinity will be a fixed “hyperbolic excess velocity”.
Rockets, though, could go 1 mph if you had enough fuel. Essentially, you’d just be hovering. The thrust that you’d apply would be slightly greater than the local force of gravity, so you’d fly around really slowly, you you wouldn’t technically be flying at “escape velocity”.
Put simply, then, escape velocity is the speed you have to be going (at the earth’s surface) to escape earth’s gravitational pull WITHOUT any additional acceleration.
To go up consistently at one mile an hour requires a constant acceleration equal to -g plus one mile per hour. Well actually, the -g part would be dropping off as you went up, but you get the idea: acceleration is constantly required until you reach the altitude where 1mph IS escape velocity.