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Amphibious drone is an Unmanned Aerial Vehicle (UAV) capable of performing in both air and under water for application in various fields, such as marine, military, and underwater habitat research, but complexity in the design and provable mathematical model to withstand the arguably swift changes in the forces during the transition phase makes it difficult to build a sustainable UAV that can operate seamlessly in both media. Beginning with the mathematical principles and physical laws, the basic concept of operation is arrived at in the present study for both media (air and water) keeping the prime objective as developing a reliable drone design and mathematical model that will satisfactorily describe the behavior of the amphibious drone with accuracy by defining the coordinate system to describe the amphibious drone's kinematics. Basic forces and torques are considered to explicitly describe the drone dynamics using Newton–Euler equations, and the final equation derived is the matrix Newton's second law. Based on the mathematical model, the final design of the drone is arrived at considering the feasibility of withstanding forces, placement of commercial-off-the-shelf components, and the amount of thrust required to carry the seamless operation. A prototype is built with active buoyancy control technique to control the underwater depth of the drone, which clearly satisfies the design and mathematical model developed in this study. [ABSTRACT FROM AUTHOR]