Most aircraft launchers exhibit a rapid acceleration of the launching aircraft, often exceeding ten times the acceleration due to gravity. However, only magnetic launchers offer flexible control over the propulsion force of the launcher cart, enabling precise control over the aircraft’s acceleration and speed during its movement on the launcher. Consequently, extensive research is being conducted on magnetic launchers to ensure the repeatability of launch parameters, protect against aircraft overloads, and ensure operator safety. This article describes the process of modeling and analyzing the dynamical properties of a launch cart of an innovative prototype launcher, which employs a passive magnetic suspension with high-temperature superconductors, developed under the GABRIEL project. The developed mathematical model of the magnetic catapult cart was employed to conduct numerical studies of the longitudinal and lateral movement of the cart, as well as the configuration of the UAV–cart system during UAV takeoff under variable atmospheric conditions. An essential aspect of the research involved experimentally determining the magnetic levitation force generated by the superconductors as a function of the gap. The results obtained demonstrate that the analyzed catapult design enables safe UAV takeoff. External factors and potential vibrations resulting from uneven mass distribution in the UAV–cart system are effectively balanced by the magnetic forces arising from the Meissner effect and the flux pinning phenomenon. The primary advantage of the magnetic levitation catapult, in comparison to commercial catapults, lies in its ability to provide a reduced and consistent acceleration throughout the entire takeoff process.