A puck of mass located at slides without friction under the influence of gravity on a surface of height . Show that the equations of motion can be approximated by
where is the gravitational acceleration and the small slope approximation is used.
Determine the motion of the puck when .
Sketch the surface
as a function of , where . Write down the equations of motion of the puck on this surface in polar coordinates under the assumption that the small slope approximation can be used. Show that , the angular momentum per unit mass about the origin, is conserved. Show also that the initial kinetic energy per unit mass of the puck is if the puck is released at radius with negligible radial velocity. Determine and sketch as a function of for this release condition. What condition relating and must be satisfied for the orbit to be bounded?