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The integral is from −∞ to ∞ , from −∞ to r plus from r to ∞ supposing both masses moved from each other mathematically.

Quote from: Yahya on 06/04/2018 18:45:16The integral is from −∞ to ∞ , from −∞ to r plus from r to ∞ supposing both masses moved from each other mathematically.This is incorrect, mathematically.The potential energy of m is measured relative to M (so M does not move)

If the original equation you speak of was wrong, we would know by now. A rocket requiring twice as much energy to reach a given altitude than originally anticipated is not something that would go unnoticed by space agencies.

the equation speaks only about escape energy, for escape speed I think no agency would use the minimum amount of it which should increase by I think √2https://en.wikipedia.org/wiki/Escape_velocity

as New Horizons has reached escape velocity; in principle it could travel to infinity, provided it didn't run into a comet or get deflected by another star.

nothing actually reaches infinity , it would be traveling with real values for distances r between the two masses M and m, my claims are in the imaginary case when r=∞.(reply #5)

I doubt this accuracy.you can't tell this for sure unless you experiment those spacecrafts traveling fly from Earth out to the distant planets or you have evident information .it is all about aerospace engineering and scientists do not have the time to test this.

Quote from: Yahya on 08/04/2018 05:52:32I doubt this accuracy.you can't tell this for sure unless you experiment those spacecrafts traveling fly from Earth out to the distant planets or you have evident information .it is all about aerospace engineering and scientists do not have the time to test this.Seriously?

OK,We know that the usual form of the equation works to very good accuracy over a range that lets us fly from Earth out to the distant planets.

Quote from: Bored chemist on 07/04/2018 12:02:09OK,We know that the usual form of the equation works to very good accuracy over a range that lets us fly from Earth out to the distant planets.escape velocity must be increased for air resistance, potential energy must be higher for energy losses and air resistance , I doubt a test for escape velocity .

You do know they also launched from the Moon, don't you?

Face it, they would have noticed if they found it took twice as much fuel or half as much as they expected.

What about this problem ?If the mass m is at infinity then the distance between them is also infinity it is logical if the distance r=∞ that both masses are at infinity, a distance to reach infinity should extend from both directions in order for it to reach infinity, if it extends from one direction the mass m will go forever and the limit can't be approached. if the distance r did not reach infinity then the mass m did not in fact reach infinity and the equation won't satisfyfrom my Op"nothing reaches infinity but when supposing that infinity is a point then the origin 0 should also be a point at minus infinity "My claim that " a distance to reach infinity should extend from both directions in order for it to reach infinity" is completely logical and simple, may someone tell me if I'm wrong?

Quote from: Yahya on 10/04/2018 09:03:51What about this problem ?If the mass m is at infinity then the distance between them is also infinity it is logical if the distance r=∞ that both masses are at infinity, a distance to reach infinity should extend from both directions in order for it to reach infinity, if it extends from one direction the mass m will go forever and the limit can't be approached. if the distance r did not reach infinity then the mass m did not in fact reach infinity and the equation won't satisfyfrom my Op"nothing reaches infinity but when supposing that infinity is a point then the origin 0 should also be a point at minus infinity "My claim that " a distance to reach infinity should extend from both directions in order for it to reach infinity" is completely logical and simple, may someone tell me if I'm wrong?The idea of infinitely is it is endless, nothing can reach infinitely.