Motion of a ball rolled over a shallow step

TL;DR

This paper derives equations describing the motion of a ball rolling over a shallow step, demonstrating experimentally how the ball's velocity and trajectory are affected, with potential classroom applications.

Contribution

It provides a mathematical model and experimental validation for the behavior of a rolling ball over a shallow step, considering non-slipping conditions.

Findings

Ball experiences velocity increase along the perpendicular direction to the step.

Derived equations accurately predict the ball's trajectory and speed changes.

Experimental results confirm the theoretical model's validity.

Abstract

A ball rolled over a shallow step will experience an increase in velocity along the direction perpendicular to the step. This causes a deflection in the ball's trajectory. In this paper we derive the equations that describe the motion of a ball rolled over a shallow step and present the results of our experimental test. This simple demonstration can be used in any classroom where the physics teacher has access to a ball and a stack of papers. Prior work has shown that a ball rolled over an edge can maintain its speed, as is commonly assumed, but it can also experience an increase or even decrease in speed. The ball can either roll without slipping while it is in contact with the edge, or else begin to slip before it leaves the edge. In this paper we will consider the case where the ball rolls without slipping the entire time it is in contact with the step edge, then contacts a lower platform. We work with shallow step heights relative to the radius of the ball so that the motion of the ball is easy to observe at all times, and so that the ball does not bounce when it encounters the lower platform. These shallow step heights mean that we can assume the ball does not slip as it moves over the edge.