Models

Hacky Sacs vary significantly in several ways. As all hacky sacs consist of a sac of some sort weighted with some sort of filling material. Materials for the sac vary from leather to hemp and from sewn panels to crocheted strands; most important though, is the rigidity of the material. To achieve a stall, the footbag and appendage must undergo an inelastic collision. The greater the amount of deformation of the footbag, the more energy is lost in altering its shape, lowering its velocity and aiding in the stall. Additionally, the material filling the footbag, which typically consists of plastic pellots or sand, affects the footbag's mass and density.

Because the footbag consists, literally, of a system of particles, it behaves slightly differently than a uniform shell like a soccer ball. An force applied perpendicular to the surface of the footbag for a short period can initiate rotation of the exterior cloth, which internal friction applies to the fill material, storing some energy as rotational kinetic energy.

Motion

Here is a model for the motion of the footbag during in a Rainbow:

In this system, the work done by the leg is not as important as the rate of work done by it. If the footbag's acceleration is altered too quickly, due to the non-planar nature of the foot (although the ideal foot pictured here appears planar), the force of friction on the foot would not be sufficient to prevent it from rolling or possibly bouncing off. To accomplish this, the initial leg position relies primarily on the angle formed between the upper leg and vertical, so changing that angle at a reasonable rate will slow the footbag's vertical motion and translate some of it into horizontal. At this point the leg is then straightened (not nescesarilly perfectly) and drawn backwards. Instead of relying primarily on the upper leg, the lower leg creates the majority of the upward motion due to the orientation of the knee joint. Interesting however, is the fact that the foot in this diagram is planar, and never capable of generating a horizontal force with the exception of one of a frictional nature (which is not sufficient to perform the trick).

For an explanation of the horizontal forces, Click Here