Particle sliding on a rough incline

We study a particle sliding on a rough inclined plane as an example of a mechanical problem with nonholonomic constraint. The particle is launched in an arbitrary direction so that its motion has both a horizontal and a 'vertical' (i.e., up- and downhill) direction. The friction force acts along the instantaneous velocity, so that the horizontal motion and vertical motion depend on each other. We discuss the character of the trajectory in the framework of dimensional analysis: the trajectory is described by a single length scale, and not by two length scales, as is the case for projectile motion. We use elementary methods to find the solutions of the equation of motion and the trajectory.