= Solution
A nearby <fundamental observer> measures radial proper distance $d\ell=a\,d\chi$ and proper time $dt$. Thus the signed radial <peculiar velocity> is
$$
\boxed{v=a\frac{d\chi}{dt}=\frac{a\dot\chi}{\dot t}.}
$$
Substitute $a^2\dot\chi=C$ and the massive normalization from part (b):
$$
\boxed{v(a)=\frac{C/a}{\sqrt{1+C^2/(a^2c^2)}},\qquad a\gamma v=C.}
$$
Equivalently, for velocity $v_0$ at scale factor $a_0$,
$$
v(a)=\frac{v_0a_0}{\sqrt{a^2(1-v_0^2/c^2)+a_0^2v_0^2/c^2}}.
$$
This is the exact law of <redshifting of peculiar momentum>. In the nonrelativistic limit, $v\propto a^{-1}$; an initially relativistic speed stays below $c$ and decreases as the universe expands.
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