Sure, you can build rockets that will escape the Earth's gravity
more slowly. The futuristic rocket you describe would have to have a
much higher exhaust velocity than current rockets have so that it can
get more thrust while burning its fuel more slowly.
It turns out that this is a waste of fuel, though. Lifting a
rocket up slowly in a gravitational field requires overcoming an extra
constant downwards thrust. If the lifting is done slowly, the force of
gravity times the time it acts is the change in momentum of the rocket
due to gravity, which must be offset by throwing burnt fuel (exhaust)
in the opposite direction. In the limit of a very very slow rocket
which just hovers over the ground, it burns fuel but does not go
It turns out the most economical use of fuel for a rocket is to
burn it all as quickly as possible and let the rocket fly away
ballistically when the fuel is gone.
It is also not possible to include an unlimited source of fuel
because the unburnt fuel must be lifted along with the rest of the
rocket. This is inescapable, because in order to get a Newton's 3rd law
reaction push, some mass has to be tossed out the back side of a
rocket, and that mass must have been carried along (of course if your
rocket acquires mass from its surroundings, like air, and throws it out
the back, you've made a jet engine which isn't a bad idea).
As for the escape velocity, that's just the the speed such that the
kinetic energy of the rocket is big enough so that as it pulls out of
the Earth's gravitational field and exchanges kinetic for potential
energy, it has enough to make it "all the way out".
(published on 10/22/2007)