  By definition, the del Pezzo $3$-fold $V_5$ is the intersection of
$\mathrm{Gr}(2,5)$ with three hyperplanes in $\mathbb{P}^9$ under the Pl\"ucker
embedding. Rational curves in $V_5$ have been studied in various contents of
Fano geometry. In this paper, we propose an explicit birational relation of the
Kontsevich and Simpson compactifications of twisted cubic curves in $V_5$. As a
direct corollary, we obtain a desingularized model of Kontsevich
compactification which induces the intersection cohomology group of
Kontsevich's space.
