Gravity, kinetic energy and distance.

[Circlotron]

Say you are on a non-rotating earth size planet with no atmosphere and no other objects with gravity exist. Totally alone in the universe. You fling a projectile away from the planet in a straight line and the faster you fling it the further it will go before it eventually falls back to the ground. Because the gravity from this planet gets asymptotically weaker the further you get from the planet, but (presumably) never actually ceases, would it be possible to fling your projectile fast enough so that it would never fall back to the ground? Or would gravity, no matter how weak, eventually win and make it come back?

[bob91343]

Since the gravitation never goes to zero, the missile will slow and then return.  The only way I can see this not happening is when you shoot it at relativistic velocity such that the equations change.

[Benta]

"Escape Velocity":
https://en.wikipedia.org/wiki/Escape_velocity#:~:text=In%20celestial%20mechanics%2C%20escape%20velocity,an%20infinite%20distance%20from%20it.

[Brumby]

You have half the answer here:

Because the gravity from this planet gets asymptotically weaker the further you get from the planet, but (presumably) never actually ceases

The other half is that the speed of the object also gets asymptotically closer to zero - where it never actually stops.  So long as this speed is large enough that the gravitational influence is insufficient to bring it to a stop, it will continue to move away.  Even if that object is moving away at 1mm per year - it is still moving away.

[Circlotron]

Okay, given that above a certain velocity an object can escape the gravity of the planet, that must mean there is a maximum amount of potential energy per kg that object can gain while moving away from the planet if it were able to fall back again. What would that constant be in kj per kg per g? Does it have a name? 

[bdunham7]

The other half is that the speed of the object also gets asymptotically closer to zero - where it never actually stops.  So long as this speed is large enough that the gravitational influence is insufficient to bring it to a stop, it will continue to move away.  Even if that object is moving away at 1mm per year - it is still moving away.

The speed approaches some speed, but not necessarily zero.  If the speed approaches zero at a distance of infinity, then the object had exactly the minimum escape velocity.

[bdunham7]

Okay, given that above a certain velocity an object can escape the gravity of the planet, that must mean there is a maximum amount of potential energy per kg that object can gain while moving away from the planet if it were able to fall back again. What would that constant be in kj per kg per g? Does it have a name?

You'd have to integrate the gravitational force over from infinity to the planet's surface.  At infinity you'd call that the object's potential energy.  At rest on the surface, I don't know of any common term for it.

[HighVoltage]

This is only a function of velocity:
- If your projectile speed is above 11.2 km/s, it will leave the earths orbit and
- If your speed is below 11.2 km/s, it will eventually crash back to the planet.