Fuzzification of a Projectile Motion Equation

Due to the extension of the concept of interval arithmetic, fuzzy numbers, which are a specific category of fuzzy sets, are often favored for representing real-world phenomena that involve uncertainty. This paper deals with the fuzzification of projectile motion, a classical physical phenomenon observed in the real world, which leads to the construction of a triangular parabolic fuzzy number, referred to as a projectile fuzzy number (PFN), whose properties are analyzed and discussed in detail. The discussion also covers the arithmetic operations involving two PFNs, not only with the established alpha-cut method but also with an approximation method, preserving the shape after operations. The fuzzification of projectile motion is utilized once more to construct a fuzzy equivalence relation, known as a projectile fuzzy equivalence relation. It has been shown that the dissimilarity of the projectile fuzzy equivalence relation constitutes a metric, the formula for which is presented explicitly. The application part of the study illustrates the qualitative fuzzy data clustering of the Iris dataset, the cancer dataset, and the crude oil dataset by employing the established projectile fuzzy equivalence relation as a similarity measure among the data points.