Show that along the optimal trajectory the quantity

is a constant. Deduce from this that along the trajectory the quantity f (t) = ˙y/x is a constant K, and express its value in terms of C, g, and α.

A skier slides down a snowy plane slope. The plane makes an angle α with respect to the horizontal direction. The skier is in the vertical field of gravity, of acceleration g. The skier starts with a zero velocity from some point O and wants to reach a given point A, downhill, in the shortest time. What is the optimal trajectory?
We choose in the plane a reference frame with origin at O, with horizontal axis Oy, and whose x axis is along the line of greatest slope, as shown in the Fig. 2.10.
We choose the origin of the potential energy at point O so that the initial energy E of the skier is zero.
We neglect friction of air and the track, as well as the efforts of the skier to maintain his trajectory. Therefore, the total energy of the skier is a constant of the motion.

Transcribed Image Text:

C = y/x(1 + (y))