http://www.aapt.org/Programs/contest...08_fnet_ma.pdf [Broken] 1. The problem statement, all variables and given/known data A uniform circular ring of radius R is ﬁxed in place. A particle is placed on the axis of the ring at a distance much greater than R and allowed to fall towards the ring under the inﬂuence of the ring’s gravity. The particle achieves a maximum speed v. The ring is replaced with one of the same (linear) mass density but radius 2R, and the experiment is repeated. What is the new maximum speed of the particle? 2. Relevant equations U = -Gm1m2 / r2 basic kinematics ρ= m/V = m / (2∏rA) (A = cross sectional area, although it is negligible I believe b/c they give you linear mass density) mring = (2∏r)ρA acceleration = Fg / mparticle = (Gmring)/d2 = (G(2∏r)ρA) / d2 3. The attempt at a solution Here's how I solved this problem TWO DIFFERENT WAYS (once with kinematics, once with conservation of energy) and got (√2)v both times. The answer is 2v. 1) kinematics asecond instance / afirst instance = a2 / a1 = [Gm2 / x2 ] / [Gm1 / x2 ] = m2 / m1 = 4∏rAρ / 2∏rAρ = 2. So a2 = 2a1. Using vf2 = 2ad, I got vf2 / vf1 = √2(2a1)d / √2a1d So I got the second instance's velocity = √2(first instance's velocity) 2) Conservation of energy U = Gm / r → U2 / U1 = ((4∏r)GAρ) / ((2∏r)GAρ) And so I got U2 = 2U1. Which means if the second instance has twice as much potential energy, it ends with twice as much kinetic energy. KE = mv2 / 2 → again I get √2 v Erhalkdjflnaeflkjafsfdlkji'msofrustrated Thanks in advance.