## Infinte distance with finite energy?

Let's say we have an object. And we then say that E = (1/2)mv2.
so we solve for velocity and say that
v = (2E/m)^(1/2)

then integrate both sides with respect to time

∫(2E/m)^(1/2)dt= ∫v dt

so we then have

(2E/m)^(1/2)t = distance

so if time was infinitely large (long) could an object travel an infinite distance if it was given in infinitesimally small unit of energy?

as i typed this out i realized that without an acceleration in obviously could but in my head this math means it's accelarating infinitely slow.

Don't hate me if this is wrong. I've only taken Calc 1 and high school physics. and I also have trouble explaining my scientific thoughts to others haha oh well. Please be indepth when denying my math. I also did substite E for (1/2)mv^2 at

∫(2E/m)^(1/2)dt= ∫v dt

this part but i got d = d^(1/2) for an answer. Thanks!
 Mentor Newton's first law tells us that this must be true.
 Hmm, it's a little difficult for me.

## Infinte distance with finite energy?

But could you accelarate the object over an infinite distance with finite energy?
 Remember that E is a function of time, at least I think it is, so the integral might not be so simple.

Mentor
 Quote by maughanster But could you accelarate the object over an infinite distance with finite energy?
Not unless it is a hyperbolicalaly decreasing acceleration -- but you can just briefly accelerate to an infinitessimally small speed and just continue at that speed forever.
 With what you just said and thinking things through a bit I think I finally figured out what i wanted to in the first place. thanks!!!
 I agree with Russ Watters, but also would like to light up some extra points. The situation of travelling forever is possible if the total energy of the body is positive. You can't just express E = 1/2 mv^2. That would be the case if you are thinking that the body is alone in the space but in real life it is in a number of fields. So, the only condition according to Newton and Kepler is taking the total energy to be positive. So, the answer is YES. A body can travel infinite distance with finite energy.