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I know that if you exceed orbital velocity, you will never fall-back to the planet. My question is not about orbits. It's about brute-force propulsion to achieve altitude. I'm using an intentionally slow velocity to help illustrate my point.

Imagine I have a rocket with very efficient fuel storage. My rocket can store enough energy to accelerate to 100kph shortly after leaving the ground, and continue to maintain that speed (100kph) for a very long period of time.

My rocket just goes straight up. It doesn't try to enter an orbit. As it leaves the atmosphere, it can throttle-back because there's no air resistance. As it continues to gain altitude in inter-planetary space, it can throttle-back even more because Earth's gravitational influence diminishes with distance. It just maintains enough throttle to continue moving away from Earth at 100kph.

At some point, Earth's gravitational influence would be moot, as other bodies (Jupiter, Sun), would gain relative influence. Eventually, far outside the solar system, even the Sun's influence would be insignificant.

My rocket never achieved escape velocity, but it sure did escape.

Assuming that my fuel supply could last long enough, and I wasn't concerned about travel time, could this method allow my rocket to "leave" without achieving escape velocity?

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    $\begingroup$ Yes it could. Escape velocity refers to a speed that you give an object at the planetary surface to allow it to escape. $\endgroup$ – Rob Jeffries May 30 '16 at 21:05
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Escape velocity is how fast you must go in order to keep moving away indefinitely without additional thrust. If the Earth were the only major body in the system, and you moved away from it at 100 km/h for 1180 years, you'd be 6.9 AU away. Since Earth escape velocity at that distance is only 100 km/h, you could then stop the engine and coast to infinity.

If we add the Sun to this oversimplified system, you could escape it by maintaining 100 km/h until you were 36 light-years away, which would take 390 million years. In the real universe, of course, the Earth's and Sun's neighbors make their spheres of influence much smaller.

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    $\begingroup$ Your answer nicely illustrates the difference of scale between Earth's gravitational influence and the Sun's. 6.9 AU vs. 2,270,000 AU. and 1180 years vs. 390,000,000 years. Wow! $\endgroup$ – user3384842 Aug 5 at 17:24
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The basic concept here, that you instead of relying on having a large enough velocity that the Earth can not pull you back in time just have a low and constant velocity, and keeps thrusting to counteract gravity instead, is not flawed by default, given your assumptions

Assuming that my fuel supply could last long enough, and I wasn't concerned about travel time, could this method allow my rocket to "leave" without achieving escape velocity?

However escape velocity decreases by distance, so even though you are travelling at only 100kph, that is eventually going to be above escape velocity, as the escape velocity goes towards zero when the distance goes towards infinity. You can not really escape a gravitational field, so the only meaningful metric is if you have enough kinetic energy to not be pulled back. That is not possible by your constraint of not having escape velocity.

In a sense though, you can escape the Earth's gravitational field, using the common approximation of sphere of influence. It defines a region for Earth's gravitational influence, and ignores it outside of it. The border of the SOI can be reached without ever going at escape velocity.

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