On the velocity maximization in downhill skiing

The dynamics of a skier performing downhill are considered. The equations of motion are utilized in a dimensionless form revealing the most important physical and physiological parameters controlling the skier speed. Several analytical results for the straight hill case are derived. The optimal shape of a hill in a vertical plane to produce maximal velocity increase between two given points is sought. The effect of the four dimensionless parameters appearing in the problem, accounting for air drag and lift, air density, ski friction, skier mass, and length and steepness of the hill, on the optimal hill shape and the final optimal velocity is studied. A comparison with the corresponding straight slope is made showing that the optimal shape may typically lead to a 50% increase in the final velocity. The variational procedure employed also yields the hill shape for minimal velocity increase between two points. The results may be used in speed prediction and hill design.