Yes. The escape speed is the minimum speed needed to escape from a given position, for ballistic motion (where gravity is the only force, so the thrust and drag are zero).

So if the object is able to thrust forever, it can escape.

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If you use a bad engine or cheap fuel the rocket will run out of propellant long before it even reaches space (which is far away from the point you considered).

A key quantity of rocket motors is the specific impulse I_sp, which is the exhaust velocity divided by the (surface) gravitational acceleration. If a rocket has an I_sp of e.g. 500 seconds, then it needs to expel 1/500 of its mass as propellant to maintain its speed (close to Earth's surface). The mass flow decreases over time so we can assume ~1000 seconds of burn time for a fully fueled rocket that just maintains its velocity. If you want to get far away from Earth in 1000 seconds (let's say a 5 times the radius of Earth, ~30000 km) then you need a speed of 30 km/s. That's above the escape velocity.

500 s I_sp is better than any rocket engine that has ever flown, and it's at the edge of what chemical fuels could potentially do under ideal conditions. If you plug in more realistic I_sp values (~250-450 s depending on the propellant and use) then it gets even worse, and this is still using extremely optimized engines. A hobby rocket motor might have an I_sp somewhere in the range of 50-100 s.

tl;dr: Yes you could in principle reach space and even escape Earth with a slow rocket, but in practice no method has the absurd engine performance this would need.

The ideal rocket equation relates the delta-V (change in velocity) a rocketship is capable of to the specific impulse (propellant efficiency) of the engine and the mass ratio (mass including propellant divide by mass of everything that's not propellant).

https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

The equation takes the logarithm of the mass ratio, so adding more fuel brings diminishing returns - this is the "tyranny of the rocket equation". Chemical rockets launching to Earth orbit have only a few percent of their launchpad mass as useful payload, with some more mass in the rocket itself, but around 90% of the launchpad mass is the fuel.

The equation doesn't depend on how quickly the fuel is burned. Thanks to orbital mechanics you are better off burning it all as fast as possible then coasting rather than throttling back and burning slowly, all else being equal.

This would be something of a miraculous "perpetual motion machine," being able to provide a force like that forever. I think it's a great question, though. To clear up the confusion, we need to ask what is meant by escape velocity, which I'll denote v_e.

This term is relevant because we can only provide a push for some finite time. Whether it's a rocket or a cannon ball, you get an impulse and then the projectile is off and moving. If we can't get it moving more than v_e once we stop pushing, then it will move in a parabola and fall back to earth. It might go *really* far, but it will still eventually come back down. If we provide v_e, then it will go up and maintain a stable orbit. It is "falling to earth at the same rate that earth is curving away from it." If, when we stop pushing, we exceed v_e, then it will be moving so fast that, even as earth's gravitational field decelerates it, the rocket has to be infinitely far away before that deceleration alone brings it to v = 0. So it will never come back to earth and has "escaped."

As a sidenote, keep in mind that the force of gravity decreases as you get farther away. So if our magic force generator truly provided a constant force, then "just barely moving" at earth's surface would translate into "rocket speeds" as it gets further away. This thing would just keep moving faster and faster. 1N fighting 0.9N isn't much, but 1N fighting 0.009N is totally different!