1. The problem statement, all variables and given/known data A casing of mass m can glide along a thin metal ring lying horizontally. It has an initial speed of v0. How long a distance s will it glide before stopping if its friction coefficient is μ? The answer is given as s = r/2μ * ln( v02 + √(v02 + g2r2)/ g * r ) The task is to provide the deductive path to reach that answer. 2. Relevant equations Centripetal force: mv2/r Gravitational force: mg Kinetic energy: mv2/2 3. The attempt at a solution The casing will stop once all the kinetic energy has turned into friction work. The friction is a sum of that caused by the gravitational force and the centripetal force. The friction work Wf is thus Wf = s * F = s * μ * ( mv2/r + mg ) Since Wf is equal to the kinetic energy we can write the following equation: s = Kinetic energy / F = (mv2/2) / (μ * ( mv2/r + mg )) This can be quite cleanly simplified into something that looks similar to the answer I am supposed to reach: s = r/2μ * ( v2/ (gr + v2) ) But this is where I run into a brick wall. I can't figure out how to integrate it properly, all the similar equations I have seen integrated seem to include arctan somewhere in the answer. I apologise any mistakes I made in the translation of the problem and in laying them out clearly, and thank you for your attention!