  Let $C$ be a curve with two smooth components and a single node. Let
$\mathcal{U}_C(r,w,\chi)$ be the moduli space of $w$-semistable classes of
depth one sheaves on $C$ having rank $r$ on both components and Euler
characteristic $\chi$. In this paper, under suitable assumptions, we produce a
projective bundle over the product of the moduli spaces of semistable vector
bundles of rank $r$ on each components and we show that it is birational to an
irreducible component of $\mathcal{U}_C(r,w,\chi)$. Then we prove the
rationality of the closed subset containing vector bundles with given fixed
determinant.
