  We show the compactly supported motive of the moduli stack of degree $n$
rational curves on the weighted projective stack $\mathcal{P}(a,b)$ is of mixed
Tate type over any base field $K$ with $\text{char}(K) \nmid a,b$ and has class
$\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}$ in the Grothendieck ring of
stacks. In particular, this improves upon the result of [HP] regarding the
arithmetic invariant of the moduli stack $\mathcal{L}_{1,12n} :=
\mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable
elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and
a marked Weierstrass section.
