I am looking for a semi-detailed description of the physics behind the brachistochrone problem. Basically, a brachistochrone is the shape of a ramp that takes the shortest time for a ball to roll down. This shape turns out to be a cycloid. I didn't believe it when I first heard about it, and I thought a ball would take the same amount of time to roll down a cycloid ramp as a straight ramp. But after building the ramps, the ball does indeed get to the bottom of the cycloid quicker. All the explanations that I've read are either too mathematical for me to follow, or just a sentence. I know that since the ball on the cycloid ramp starts at a steeper incline, it starts out faster, and thus reaches the bottom first, even though both balls have the same velocity at the bottom. I am still having a hard time grasping this, and I was wondering if anyone could provide a slightly more in-depth description of the physics of this situation.