Particle decay, Oberth effect and a relativistic rocket in the Schwarzschild background

We relate the known Oberth effect and nonrelativistic analogue of Penrose process. When a particle decays to two fragments, we derive conditions on angles under which debris can come out for such process occur. also consider decay in relativistic case, when moves background Schwarzschild black hole. This models rocket ejects fuel. Different scenarios are analyzed depending what data fixed. The efficiency is found, particular, near horizon photon (when ejected massless). prove directly that most efficient occurs fuel along trajectory. this horizon, reaches 100% rocket. compare ways how reverse its direction motion hole event by restoring initial energy-to-mass ratio: (i) single ejection or (ii) two-step it stops back afterwards. For nonphotonic rocket, case larger mass be taken from vicinity horizon. photonic one, there no difference between respect. briefly scenario hangs over due continuous Then, exponentially with proper time.