In this paper a new moving-boundary algorithm is presented for the solution of two-dimensional flows on unstructured grids. Application of this algorithm is investigated by simulating the internal ballistics of solid rocket motors (SRM). Solution domain is discretised to a number of triangular elements using control-volume based finite-element (CVFE) method. Euler equations are integrated over the control volumes, and Roe's flux-difference scheme is used to evaluate flux vectors at the control-volume surfaces. To eliminate numerical errors due to grid movement, geometric conservation laws (GCL) are carefully implemented. The present algorithm recognises locations along the moving boundaries where the stretched elements result in a distorted grid. To avoid this, new elements are inserted through a local re-triangulation of elements in these regions. Internal ballistics of three SRMs with cylindrical grains burning from inside, are numerically simulated. Using a heat transfer model, erosive burning is included in calculation of propellant burning rate. The aftend of third SRM is tapered to demonstrate capability of present algorithm in solving problems with moving boundaries approaching each other. Very good agreement is observed between the time variations of head-end chamber pressure, obtained from the present work and the other methods.