  Suppose that $X$ is a smooth projective variety and that $C$ is a general
member of a family of free rational curves on $X$. We prove several statements
showing that the Harder-Narasimhan filtration of $T_{X}|_{C}$ is approximately
the same as the restriction of the Harder-Narasimhan filtration of $T_{X}$ with
respect to the class of $C$. When $X$ is a Fano variety, we prove that the set
of all restricted tangent bundles for general free rational curves is
controlled by a finite set of data. We then apply our work to analyze Peyre's
"freeness" formulation of Manin's Conjecture in the setting of rational curves.
