# Finite velocity at infinite distance

If an object were launched exactly at escape velocity it would have zero velocity at infinity. But what if we launch an identical objects at greater than escape velocity? Apparently it will have a finite velocity at infinity. This is a bit puzzling. After all, if the object has a finite velocity then it is still moving away from us. If this is the case then apparently it hasn't in fact reached infinity. Is infinity different for the two objects? More likely I'm just missing something here.

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$\infty+v\Delta t=\infty$ –  fffred Jun 3 '13 at 21:40
Infinity is not an actual point. Something cannot "reach inifinity". –  jkej Jun 3 '13 at 21:50
@jkej Although I agree with your second statement, you should be careful in asserting that infinity is not a point. See en.wikipedia.org/wiki/Point_at_infinity –  joshphysics Jun 3 '13 at 22:34
@joshphysics Point taken. I shouldn't have overreached. –  jkej Jun 3 '13 at 22:47

I think your confusion here stems from not being careful about the definition of "reaching infinity" and "velocity at infinity." The punchline of the answer is in italics at the end, but here are some details:

For simplicity, let us restrict the discussion to that of a small object of mass $m$ being fired radially outward from the surface of a spherical planet of mass $M$ of uniform density.

When we fire an object radially outward with some initial speed $v_0$ and let $r(t)$ be its radial position as a function of time, then we say that the object reaches infinity provided given any distance $d>0$, there is a time $t>0$ for which $r(t)>d$.

It can be shown (using energy conservation) that there is a velocity $v_e$ which we call escape velocity such that if the object reaches infinity, then its initial velocity must satisfy $v_0\geq v_e$ and that $\dot r(t)\to 0$ as $t\to\infty$ if $v_0 = v_e$

It can also be shown (again using energy conservation) that if $v_0>v_e$, then $$\lim_{t\to\infty} \dot r(t)>0$$ In other words, if you fire the object at greater than escape velocity then not only does it reach infinity in the sense defined above, but also its velocity does not have vanishing limiting value as it goes off to infinity but instead approaches some nonzero, finite limiting value called the velocity at infinity.

Notice, in particular, that in both the $v_0 = v_e$ and $v_0>v_e$ cases, the object never actually gets infinitely far away from the planet in finite time. Reaching infinity is defined in terms of a limit.

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You should be thinking about this as an object asymptotically approaching a limiting speed (zero or otherwise) as the distance grows without bound.

The asymptote is the speed "at infinity" but do keep in mind the following: no matter how far away the object is, the distance is always finite, i.e, the object never actually reaches the limiting speed.

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