rolling down a slope (a cylinder is easier to handle than a ball): how fast will it accelerate?
If energy is conserved, after it has descended 1 meter, it will have gained the same amount of energy as an object of the same weight falling from the height of one meter. If it were just SLIDING down smoothly, with no friction, that cylinder would indeed have the same speed v as an object falling from that height. But if it rolls, some of the energy also goes to the motion of rotation around the axis of the cylinder. As a result the forward motion is slower than v, it gets only part of the energy. Calculating the energy of a solid cylinder of radius (say) R, rotating around its axis with (say) 1 revolution per second, is not easy. Different parts move at different speeds–those on the edge are fastest, with some velocity we can call V, while the middle does not move at all. What we need is find some average velocity V’, equal to some fraction of V, so we can use the formula for kinetic energy (1/2)m V’2 (one half mass times velocity V’ squared). Calculating that average takes a branch
