Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities and RT/RM interfacial mixing are omnipresent in nature and technology and are long-standing challenges in mathematics. This work reports the first rigorous derivation of the buoyancy and drag values from the conservation laws and the boundary value and initial value problems governing RT/RM dynamics with variable acceleration having power-law time dependence. Within group theory approach, we directly link the governing equations, through the dynamical system based on space groups, to the momentum model based on scaling transformations. We precisely derive the model parameters — the buoyancy and the drag, exactly integrate the model equations, and find asymptotic behaviors of the solutions for RT/RM bubbles and spikes in the linear, nonlinear and mixing regimes. The analysis provides new extensive benchmarks for future research.