A polarizable interatomic potential is developed for atomistic simulations of molten MNO3 (M = Li, Na, K) salts. The potential is parametrized using a force matching method relying on the adjustment of parameters such that density functional theory generated forces, stress tensors, and dipole moments are reproduced. Simulations conducted using the new potential are used to estimate physical parameters of the melt, which are then compared with available experimental results. The average calculated densities of NaNO3 and KNO3 are within 2% of the experimental value within the temperature range studied, while that of LiNO3 is within 3%. Thermal conductivities and viscosities are estimated using equilibrium calculations and the Green-Kubo method. The thermal conductivity values of NaNO3 and KNO3 are found to match well with experimental data, while that of LiNO3 is approximately 20% larger than experimentally determined values throughout the temperature ranges simulated. The calculated viscosities are also in good agreement with experimentally determined values. The (NaxK1-x)NO3 mixture is also investigated, with densities, thermal conductivities, and viscosities determined and compared with experimentally determined values where available. Additionally, radial and angular distribution function data is presented for all salts, revealing details of the atom-level structures present in the melts. We have found that the new interatomic potential is effective for atom scale modeling of the physical properties of molten nitrate salts.