Constant Speed Ramps

TL;DR

This paper characterizes all possible ramps allowing an object to move at constant speed under gravity and friction, providing explicit descriptions for planar ramps and a classification for space ramps based on tangent vector fields.

Contribution

It introduces a comprehensive mathematical framework for describing all constant speed ramps in both planar and spatial settings, linking solutions to tangent vector fields.

Findings

Planar ramps are described by curves with velocity vectors involving tanh and sech functions.

The number of spatial ramps corresponds to tangent unit vector fields in the south hemisphere.

Explicit formulas and visual explanations are provided for these ramps.

Abstract

In this paper we show all possible ramps where an object can move with constant speed under the effect of gravity and friction. The planar ramp are very easy to describe, just rotate a curve with velocity vector (tanh(as),sech(as)). Recall that tanh(as)^2+sech^2(as) = 1. Therefore, the solution of the planar constant speed problem is connected with easy to describe examples of curves with arc-length parameter. For ramps in the space, we show that there are as many ramps as tangent unit vector fields in the south hemisphere. A video explaining these results can be found at http://www.youtube.com/watch?v=iBrvbb0efVk