  Scattering processes are a fundamental way of experimentally probing
distributions and properties of systems in several areas of physics.
Considering two-body scattering at low energies, when the de Broglie wavelength
is larger than the range of the potential, partial waves with high angular
momentum are typically unimportant. The dominant contribution comes from $l=0$
partial waves, commonly known as $s$-wave scattering. This situation is very
relevant in atomic physics, e.g. cold atomic gases, and nuclear physics, e.g.
nuclear structure and matter. This manuscript is intended as a pedagogical
introduction to the topic while covering a numerical approach to compute the
desired quantities. We introduce low-energy scattering with particular
attention to the concepts of scattering length and effective range. These two
quantities appear in the effective-range approximation, which universally
describes low-energy processes. We outline a numerical procedure for
calculating the scattering length and effective range of spherically symmetric
two-body potentials. As examples, we apply the method to the spherical well,
modified P\"oschl-Teller, Gaussian, and Lennard-Jones potentials. We hope to
provide the tools so students can implement similar calculations and extend
them to other potentials.
