I just wanted to check my understanding of escape velocity. If a projectile was to launch and have the exact velocity as the escape velocity of the earth, it would have a final velocity of 0 correct? And this final velocity would be approached as the time approaches infinity correct?
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$\begingroup$ yes. in a situation where there are no other planets and Newton's laws holds to arbitrary large scales. $\endgroup$– PraharCommented Dec 6, 2021 at 4:30
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1$\begingroup$ @Prahar Please post answers as answers, not as comments. $\endgroup$– BioPhysicistCommented Dec 6, 2021 at 4:36
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It would have a final velocity of 0 correct?
Yes, correct.
And this final velocity would be approached as the time approaches infinity correct?
Yes, also correct.
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Yes.
If you take the only planet earth as a mass point, you will find that its orbit is a branch of parabola, and its focus is the earth itself, according to the Newton's laws.
And when times approaches infinity, the distance from the earth and rocket approaches infinity as well. So according to the work-energy theorem, its final velocity is 0. It is because there always exists $V+T=0$, where $V$ is the potential energy and $T$ is the kinetic energy. When approaching to infinity, $V\propto -\frac{1}{r}$ approaches to 0 so $T$ approaches to 0 at the same time.
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Your understanding is correct, if we make the assumption that the Earth is the only object with mass (i.e. there is no Sun, no Milky Way, no Virgo Cluster, etc).
Note that the often-quoted escape velocity of the Earth, ~11 km/s, does not actually suffice to escape the Solar System since you need a faster speed to escape the Sun's gravity.
