  We completely describe the components of expected dimension of the Hilbert
Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm
SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a
curve $C$. We show that for every $k \geq 1$ there are ${\rm gcd}(r, \deg L)$
unobstructed components. In addition, if $k$ is divisible by $r_1(r-r_1)(g-1)$
for $1\le r_1\le r-1$, there is an additional obstructed component of the
expected dimension for each such $r_1$. We construct families of obstructed
components and show that their generic point is not the generic vector bundle
of given rank and determinant. Finally, we also obtain an upper bound on the
degree of rational connectedness of ${\rm SU}_{C}(r,L)$ which is linear in the
dimension.
