In this study, the unsteady natural convection boundary-layer flow along an impulsively heated vertical isothermal plate immersed in a stably stratified semi-infinite ambient fluid is explored using scaling analysis and direct numerical simulation. Scaling relations are obtained for the thermal and velocity boundary layer thicknesses, the boundary layer velocity, the development time and the Nusselt number, in terms of the Rayleigh and Prandtl numbers and the stratification parameter. The scaling results are validated using the numerical simulations.