  Ein, Niu and Park showed in [ENP20] that if the degree of the line bundle $L$
on a curve of genus $g$ is at least $2g+2k+1$, the $k$-th secant variety of the
curve via the embedding defined by the complete linear system of $L$ is normal,
projectively normal and arithmetically Cohen-Macaulay, and they also proved
some vanishing of the Betti diagrams. However, the length of the linear strand
of weight $k+1$ of the resolution of the secant variety $\Sigma_k$ of a curve
of $g\geq2$ is still mysterious. In this paper we calculate the complete Betti
diagrams of the secant varieties of curves of genus $2$ using
Boij-S\"{o}derberg theory. The main idea is to find the pure diagrams that
contribute to the Betti diagram of the secant variety via calculating some
special positions of the Betti diagram.
