An analysis is performed to study the transient natural convection flow and heat transfer characteristics of an incompressible viscous fluid past an impulsively started semi-infinite vertical plate in which the variable surface heat flux varies as the power of the axial coordinate in the form The dimensionless governing equations here are unsteady, two-dimensional, coupled and non-linear partial differential equations. A finite-difference scheme of Crank-Nicolson type is employed to solve the problem. Velocity and temperature profiles are compared with available theoretical solution in the literature and are found to be in good agreement. The velocity and temperature profiles for different values of Prandtl number, Grashof number and n are studied. Local as well as average skin-friction and Nusselt number are shown graphically. It is observed that an increase in the value of the exponent n enhances the surface heat transfer rate.