An original Immersed Boundary Method for solving moving body flows is proposed. This method couples i) a Lagrangian Volume-of-Solid description of the solid object avoiding conservation issues and ii) a robust implicit volume penalty forcing embedded in a low-Mach number projection method to account for the solid's impact on the fluid dynamics. A new composite velocity field is introduced to describe both solid and fluid domains in a single set of governing equations. The accuracy of the method has been assessed on several academic cases, involving stationary or moving bodies and with different mesh resolutions. The predicted forces on the solid are in excellent agreement with body-fitted reference cases. The system of equations is also proven to be fully mass conservative. Application of the method on a two-dimensional vertical axis turbine case shows a 30% reduction in computational cost compared to a body-fitted method.