Problems of maximal mean resistance on the plane

A two-dimensional body moves through a rarefied medium; the collisions of the medium particles with the body are absolutely elastic. The body performs both translational and slow rotational motion. It is required to select the body, from a given class of bodies, such that the average force of resistance of the medium to its motion is maximal. Numerical and analytical results concerning this problem are presented. In particular, the maximum resistance in the class of bodies contained in a convex body K is proved to be 1.5 times the resistance of K. The maximum is attained on a sequence of bodies with a very complicated boundary. The numerical study was made for somewhat more restricted classes of bodies. The obtained values of resistance are slightly lower, but the boundary of obtained bodies is much simpler, as compared with the analytical solutions.