Impulsive encounters between astrophysical objects are usually treated using the distant tide approximation (DTA) for which the impact parameter, b, is assumed to be significantly larger than the characteristic radii of the subject, rs, and the perturber, rP. The perturber potential is then expanded as a multipole series and truncated at the quadrupole term. When the perturber is more extended than the subject, this standard approach can be extended to the case where rS ≪ b < rP. However, for encounters with b of order rs or smaller, the DTA typically overpredicts the impulse, Δbv, and hence the internal energy change of the subject, ΔEint. This is unfortunate, as these close encounters are, from an astrophysical point of view, the most interesting, potentially leading to tidal capture, mass stripping, or tidal disruption. Another drawback of the DTA is that ΔEint is proportional to the moment of inertia, which diverges unless the subject is truncated or has a density profile that falls off faster than r-5. To overcome these shortcomings, we present a fully general, non-perturbative treatment of impulsive encounters which is valid for any impact parameter, and not hampered by divergence issues, thereby negating the necessity to truncate the subject. We present analytical expressions for Δbv for a variety of perturber profiles, and apply our formalism to both straight-path encounters and eccentric orbits. In particular, we discuss two cases of gravitational encounters: 1. encounters between identical cold dark matter haloes, and 2. tidal shocking of satellite galaxies orbiting in a host galaxy potential. We show that predictions about mass loss and energy injected by tidal shocks from our non-perturbative formalism are in much better agreement with numerical simulations than the DTA. We infer that our non-perturbative formalism should be used instead of the DTA in treatments of tidal disruption and mergers, e.g. the calculation of galaxy merger-rates.