The point of departure of a particle sliding on a curved surface

A particle is thrown tangentially on a surface. It is shown that for some surfaces and for special initial velocities the thrown particle leaves immediately the surface, and for special conditions it never leaves the surface. The conditions for leaving the surface is investigated. The problem is studied for a surface with the cross section $y=f(x)$. The surfaces with the equations $f(x)= -\alpha x^{k}\ (\alpha, k>0)$ is considered in more detail. At the end the effect of friction is also considered.