This paper discusses the development of a simple model for describing the decay of a homogeneous turbulence subjected to a weak fluid acceleration. A sufficiently weak fluid acceleration may not affect the anisotropy of a decaying homogeneous turbulence. Additionally, the turbulent kinetic energy and its dissipation each follow a power law in homogeneous turbulence when the weak fluid acceleration is absent. Considering these characteristics, the governing equations describing the effect of the weak acceleration on the turbulent kinetic energy and its dissipation are derived. The derived equations are then numerically simulated. When the fluid acceleration is sufficiently small, its effect on the turbulent time scale is negligible. This result simplifies the governing equation for the kinetic energy and yields a simple formula describing the effect of the fluid acceleration. However, the derived simple formula does not necessarily agree with the numerical results. The deviation between the simple formula and the numerical results is considered to be characterized by the decay exponent.