An elementary method for calculating the duration of motion of a block with mass attached to the end of a spring under the influence of kinetic friction

Abstract

Calculating time intervals in motions with non-constant acceleration is a challenging task. In most cases, advanced mathematical tools are required. Therefore, these topics are usually not suitable for presentation at the high school level. In this paper, the case of the motion of a block of mass attached to the end of a linear spring on a rough surface is revisited. The goal is to find an elementary method for calculating the time interval required to cover a distance between two arbitrary positions suitable for teaching at the high school level. To accomplish this goal, a mathematically ‘equivalent problem’ involving the combination of simple harmonic motions between two different equilibrium positions is considered. It effectively conveys that with this approach, the calculation of time intervals relies solely on simple trigonometry. In addition, students can draw interesting conclusions, such as the linear decay of the oscillation’s amplitude and the independence of the period from the oscillation’s amplitude.