Abstract: Variational quantum algorithms (VQAs) are hybrid quantum–classical algorithms that seek to harness the advantage of quantum computers while simultaneously mitigating the drawbacks of the noisy, intermediate-scale (NISQ) quantum hardware existing today. VQAs have an established theoretical potential, but their ability to effectively solve problems arising from practical applications, and whether this utility can be wholly replicated by quantum-inspired classical algorithms, remains an active area of interest. We present a novel application of both the Variational Quantum Linear Solver (VQLS) and the Variational Neural Linear Solver (VNLS)—an existing VQA for solving systems of linear equations, and its quantum-inspired fully classical counterpart—as the key component within a larger minimum map Newton solver for a complementarity-based rigid body contact model. Using each algorithm, we demonstrate that this solver accurately depicts the dynamics of the model system’s rigid spherical bodies as they collide. These results indicate that quantum and quantum-inspired linear algebra algorithms may provide a satisfactory alternative to standard linear algebra solvers for modeling certain physical systems.