Publication Date

Spring 5-5-2026

Document Type

Student Project

First Advisor

Deng, Sien

Degree Name

B.S. (Bachelor of Science)

Department

Department of Mathematical Sciences

Abstract

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling ladder is predicted to hit the ground with infinite acceleration, the bobber is predicted to move horizontally with infinite acceleration as it reaches the dock, the melting snowball’s radius shrinks at an unbounded rate as it approaches zero, and the water level in a conical tank is predicted to fall infinitely fast as the tank empties. These results are physically impossible and indicate a breakdown in the mathematical model rather than an error in calculus. The issue is that the classical related-rates approach assumes a single model remains valid throughout the entire motion, even near physical boundaries where the assumptions of the model no longer hold. This paper focuses on the fishing bobber problem and introduces a two-phase modeling approach. The standard related-rates model is shown to be valid only up to a critical distance. Beyond this point, a different physical model governs the motion. This framework removes the unphysical infinities and produces results that are consistent with basic physical principles. To the authors’ knowledge, no standard calculus text or published analysis explicitly identifies or resolves this transition using a two-phase dynamical framework. Although the bobber problem is the main example, the ideas apply more broadly to other classical related-rates problems found in the calculus literature.

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