1. The problem statement, all variables and given/known data A solid homogeneous cylinder of mass M and radius R is moving on a surface with a coefficient of kinetic friction μk. At t=0 the motion f the cylinder is purely translational with a velocity v0 that is parallel to the surface and perpendicular to the central axis of the cylinder. Determine the time tR after which the cylinder performs pure rolling motion. 2. Relevant equations None were given on the paper but I assume I'll be needing this: When rolling: ƩE = K = 0.5mv2 + 0.5Icmω2 3. The attempt at a solution Here's what I've worked through so far When the cylinder is in pure translational motion: ƩE = 0.5Mv02 - fkd ƩE = 0.5Mv02 -μknd When the cylinder is in pure rolling motion: ƩE = K = 0.5MvR2 + 0.5Icmω2 Due to conservation of energy: 0.5Mv02 - μknd = 0.5MvR2 + 0.5Icmω2 As n = -Mg and Icm = 0.5mr2 0.5Mv02 + μkMgd = 0.5MvR2 + 0.25MR2ω2 Simplifying: 0.5v02 + 9.8μkd = 0.5vR2 + 0.25R2ω2 As vcm = rω 0.5v02 + 9.8μkd = 0.5vR2 + 0.25vR2 0.5v02 + 9.8μkd = 0.75vR2 After this I can't really figure out how to go on apart from substituting d for: 0.5(v0 + vR)t Which would add a time variable in. This gives: 0.5v02 + 4.9μk(v0 + vR)t = 0.75vR2 The main things I'm stuck on are whether or not the vcm at t=t0 (v0) and the vcm at t=tR (vR) are the same or not and how to get rid of the coefficient of kinetic friction which I am not given a value for.