A new extended metric space and solution of rocket’s ascending motion

In this work, we introduce extended [Formula: see text]-metric space and extended [Formula: see text]-metric space with a focus on their topological notions including relations between [Formula: see text]-metric, extended [Formula: see text]-metric and extended [Formula: see text]-metric. By challenging the conventional assumption of zero self-distance, we investigate novel postulates to establish the existence and uniqueness of fixed point theorems in these domains and pave the way for more accurate mathematical models applicable to real-world scenarios. Our findings are supported by illustrative examples. Additionally, using our result we explore the problem that indicates the ascending motion of a rocket. Consequently, our research contributes not only to a deeper comprehension of mathematical concepts but also to practical utility across various scientific domains.