# Kinematics of an Object Sliding along Terrain

## Main Question or Discussion Point

Hello,

I need help understanding the kinematics of an object sliding along a surface (not just a plane, but a changing surface), as I cannot find any information on this topic. Let's assume I have an object on a frictionless smooth continuous/differentiable surface in the field of gravity and that energy is conserved.

If this surface is just an inclined plane, then the motion of the object is described simply by the projection of gravity on the plane. However, what if the slope then smoothly inclines upward? Experience tells us that the object's path of motion will adhere to the shape of the surface and furthermore the shape does not affect |v| (only g*h does).

This cannot be described in terms of gravity/normal forces, so I assume the reason this happen is that an elastic collision occurs between the object and the surface and since the object's velocity was originally tangential to the surface, the reflected velocity will remain tangential (as long as the surface is differentiable).

The question now becomes what happens if the surface slopes down. In the case of a ramp, experience tells me if an object is going sufficiently fast it will simply jump off the surface and become a projectile. If it is not going fast enough, it will remain on the surface.

However, the math in this case would show that no matter the speed of the object, at the moment the path slopes down, it will have an upward velocity component and leave the path. Since the object is no longer tangential to the path, upon re-entry, it will start bouncing and this is what I get when I try to write a simulation: the object will at first nicely slide up/down a path, make the predicted jump and then start to bounce and never stop bouncing.

The scenario I'm looking to emulate is the classic one of the penguin/skier sliding on ice. The skier should jump off the surface when going sufficiently fast at an incline and return to the surface after doing a jump.

It is obvious that collision with the surface is not elastic in this case as the skier does not bounce, but then the velocity component along the surface should be attenuated likewise and motion would be short lived when the path changes shape. What's going on here? Also, there's the related problem of a wheel rolling on a surface with infinite friction.