# Preliminary Calculations for a Mathematical Model of the Behavior of a Dog "Fetching" a Ball in Water

## Institution

Kentucky Community and Technical College System

## Abstract

It can be shown that over time, a dog learns to retrieve an object in water in the minimum amount of time by using an optimal amount of running on land before it jumps into the water. In a field experiment, a ball was thrown into a pond at varying distances and angles from the edge, and a Labrador retriever, Hank, was commanded to "fetch" the ball. The behavior of running and swimming to retrieve the ball and bring it back to his master constituted a trial. Observers measured angles, distance running on land, and time running and swimming on each trial. Trigonometry was used to obtain the distance swimming on each trial. Calculus was used to obtain the minimum total time to reach the ball and optimal point of entry into the water, as a function of swimming and running rates. The percent differences in total time and the optimal time, and distance running and optimal distance running on each trial were graphed. The curve from preliminary trials looks like a curve of "trial and error" learning.

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Preliminary Calculations for a Mathematical Model of the Behavior of a Dog "Fetching" a Ball in Water

It can be shown that over time, a dog learns to retrieve an object in water in the minimum amount of time by using an optimal amount of running on land before it jumps into the water. In a field experiment, a ball was thrown into a pond at varying distances and angles from the edge, and a Labrador retriever, Hank, was commanded to "fetch" the ball. The behavior of running and swimming to retrieve the ball and bring it back to his master constituted a trial. Observers measured angles, distance running on land, and time running and swimming on each trial. Trigonometry was used to obtain the distance swimming on each trial. Calculus was used to obtain the minimum total time to reach the ball and optimal point of entry into the water, as a function of swimming and running rates. The percent differences in total time and the optimal time, and distance running and optimal distance running on each trial were graphed. The curve from preliminary trials looks like a curve of "trial and error" learning.