What's Wrong with Vacuum-Filled Vertical Tube Escape Velocity?

[a1call]

Ignoring drag, terminal velocity and friction, input 1143 seconds here:
http://keisan.casio.com/exec/system/1224835316
It yields a velocity of 11.2 km/s (Earth's Escape Velocity) at freefall from a height of 6.4 km (a fraction of the height/depth of Everest, Antarctic ice cap and deepest oceanic depth).
Consider a 6.5 km Vaccum filled vertical tube with a half circle at the bottom which a ball could roll at the bottom and redirect upwards after freefall through it.
Such a ball will have a velocity greater than the escape velocity at the bottom of the tube.
This can't be right since considering the tube be a complete U shape then at top/end of the path the speed would be the same as the start of the path namely 0 and no escape would be achieved.
Where did I go wrong?
Thanks in advance.

 

[Bandersnatch]

Where did I go wrong?

Here:

6.4 km

Count the digits. You're off by a factor of 1000.

Also, keep in mind that the equation used there is an approximation for a uniform gravitational field (near the surface only). That g is not actually a constant, so the higher you go the more wrong your numbers will be.

 

[a1call]

Yes thank you I saw my error and was also pointed out on the other board that gravity decreases by altitude.
But I am still confused.
The question now is that changing the height of the tube to whatever value where we could achieve escape velocity at the bottom would still be a conflict from 0 velocity at the end of the U path. Unless the required height happens to be infinity. Is it?

 

[Bandersnatch]

Is it?

It is.
Check the definition of escape velocity. Usually it's done in terms of kinetic energy equalling potential energy. At infinity the body has got the maximum potential energy (least negative, so 0) and minimum kinetic energy (0). It follows that the velocity is 0 there.

In other words, escape velocity is the velocity a body needs to be able to fully climb out of the gravity well of some other massive body with no leftover velocity. And since gravity extends to infinity, the potential is 0 only there.

 

[Nugatory]

Unless the required height happens to be infinity. Is it?

Yes, that's the definition of escape velocity - the lowest speed at which the object will never return to earth. If the speed were to reach zero at any finite height the object would eventually fall back to Earth (this is the exact same situation as if we held the object at rest at that height than released it - it would fall).

 

[a1call]

Thank you both for resolving my issue.

 

[mac_alleb]

Seems like 4g enough.