The elastic scattering of positive kaons from $^{12}\mathrm{C}$, $^{3}\mathrm{He}$, $^{3}\mathrm{H}$, and $^{4}\mathrm{He}$ is calculated with a theoretical momentum space optical potential. The theory includes nuclear spin, nucleon recoil and binding, a Lorentz invariant angle transformation, realistic nuclear form factors for the proton and neutron matter and spin distributions, and a kaon-nucleon $T$ matrix with off-shell behavior based on a separable potential model. Differential and total cross sections, polarizations, and isotopic ratios are examined for kaon energies from 0.4 to 1 GeV and compared with results from pion and electron scattering.NUCLEAR REACTIONS $^{12}\mathrm{C}$(${K}^{+}$,${K}^{+}$), $^{4}\mathrm{He}$(${K}^{+}$,${K}^{+}$), $^{3}\mathrm{He}$(${K}^{+}$,${K}^{+}$), $^{3}\mathrm{H}$(${K}^{+}$,${K}^{+}$), $^{3}\mathrm{He}$(${K}^{0}$,${K}^{0}$); $E=39\ensuremath{-}804$ MeV; $\ensuremath{\sigma}(\ensuremath{\theta})$ and ${\ensuremath{\sigma}}_{\mathrm{tot}}$; theoretical calculation, momentum space optical potential; spin effects, binding, recoil, angle transformation; compare with $^{12}\mathrm{C}$ data.