EDIT: This is in the wrong section isn't it? How do I move it to the General Physics section? (My bad.) This isn't a homework problem (I'm not in a physics class) so hopefully this isn't the wrong section. My question is about finding the minimum velocity needed for an object (directly upwards) to never return to Earth. (**Ignoring air resistance and other subtleties such as other sources of gravity and such**) I have an idea for finding it, but I've gotten stuck. My idea was to make it "choppy" (turn acceleration/velocity into being piece-wise-linear) and then take the limits as the piece size gets infinitely small. I've turned it into an equation that I think would solve it, but the problem is that it's an infinite sum and I can't make progress towards evaluating it because I can't figure out how to generalize the nth term in terms of n. Perhaps that makes this a bit more of a math problem, (but most physics is, to an extent,) so sorry if this is in the wrong section (this is my first post). I've posted a picture of my "solution" (I don't know where it is but I'll assume it will appear). The equation in the red circle (at the top) is my "solution" G is the universal constant of gravity, M is the mass of Earth, r is the radius of Earth and V is the great thing I'm trying to solve for. My idea behind this equation: b(subscript)1 is the change in velocity over the first height interval (Δh) because Δh divided by v is the time spent moving through that interval, and a(subscript)1 is the acceleration at that height. Then for the next height interval you replace v with v+change-in-v (in other words, replace v with v+b(subscript)1) and apply the same logic (and so on for all terms). Taking the infinite sum of this would give you the change in velocity as you got infinitely far. Taking the limit as Δh→0 would make it "more true" (because acceleration is continuous, not "choppy") and setting it equal to -v would mean you only reach zero velocity at an "infinite distance" (and that would therefore make v the minimum velocity, right?) My questions are: 1.) Is the logic/equation correct? Would it give you the appropriate minimum velocity? (I think it would, but then again, I ALWAYS think my logic is right, and it very often isn't.) 2.) Is the equation solvable? Is this an adequate approach to the problem? Can you solve it for v? (I think you can, since there's only constants and v, but the infinite sum might cause problems, I'm not very mathematically advanced so I don't know) 3.) Is it possible to generalize b(subscript)n in terms of n? Is it possible to evaluate the sum without generalizing it in terms of n? (If so how?) Are there other methods of evaluating the sum? Question 3.) is the "wall" that i've run into. I can't solve the problem because I can't figure out how to generalize b(subscript)n in terms of n. (I've generalized it in the only concise way I know how, but I don't think it's solvable like as it is. It would take some genius maths I think.) Can anyone help with these questions? Thank you and sorry for the lengthy post.