Experimento didático com a braquistócrona e outras curvas

Abstract

In this work, a didactic experiment suitable for high school students will be built, introducing the content in such a way that its curiosity is instigated with methodological guidelines capable of making learning more pleasant and interactive. In this experiment we will build: a ramp in the shape of a line, a parabola and a brachistochrone. The objective is to show which of the trajectories traveled by two furniture, marble and one polished block, in the vertical plane, achieves the shortest possible time between two pre-fixed points and by the action of local gravity. The theoretical basis behind minimizing the time on a particle trajectory between two points in the vertical plane is the variational calculation and the Fermat principle, from which the Euler-Lagrange equations of motion are obtained, which is equivalent to the equations of motion Newton. For our specific case, the solutions of the Euler-Lagrange equations show that the path (curve) that minimizes the particle travel time turns out to be an inverted cycloid curve, which in this case is called brachistochrone. The trajectory of a particle is called brachistochrone, which is subjected to a constant gravitational field, without friction, with zero initial velocity and each time it rolls along the path it changes its final velocity. We intend, with the elaboration of this thesis, to show that the study of curves and their properties such as their speed and time is relevant in basic education. We believe that it can be presented as curiosity and analysis by students.