We consider a finite element approximation for a system consisting of the evolution
of a closed planar curve by forced curve shortening flow coupled to a reaction-diffusion equation on
the evolving curve. The scheme for the curve evolution is based on a parametric description allowing
for tangential motion, whereas the discretisation for the PDE on the curve uses an idea from [6].
We prove optimal error bounds for the resulting fully discrete approximation and present numerical
experiments. These confirm our estimates and also illustrate the advantage of the tangential motion
of the mesh points in practice.