  In this paper we obtain a formula for the number of rational degree d curves
in $\mathbb{P}^3$ having a cusp, whose image lies in a $\mathbb{P}^2$ and that
passes through $r$ lines and $s$ points (where $r + 2s = 3d + 1$). This problem
can be viewed as a family version of the classical question of counting
rational cuspidal curves in $\mathbb{P}^2$, which has been studied earlier by
Z. Ran, R. Pandharipande and A. Zinger. We obtain this number by computing the
Euler class of a relevant bundle and then finding out the corresponding
degenerate contribution to the Euler class. The method we use is closely based
on the method followed by A. Zinger and I. Biswas, S. D'Mello, R. Mukherjee and
V. Pingali. We also verify that our answer for the characteristic numbers of
rational cuspidal planar cubics and quartics is consistent with the answer
obtained by N. Das and the first author, where they compute the characteristic
number of $\delta$-nodal planar curves in $\mathbb{P}^3$ with one cusp (for
$\delta \leq 2$).
