  Let $\mathcal{G}$ be the planar Galilean conformal algebra and
$\widetilde{\mathcal{G}}$ be its universal central extension. Then
$\mathcal{G}$ (resp. $\widetilde{\mathcal{G}}$) admits a triangular
decomposition:
$\mathcal{G}=\mathcal{G}^{+}\oplus\mathcal{G}^{0}\oplus\mathcal{G}^{-}$ (resp.
$\widetilde{\mathcal{G}}=\widetilde{\mathcal{G}}^{+}\oplus\widetilde{\mathcal{G}}^{0}\oplus\widetilde{\mathcal{G}}^{-}$).
In this paper, we study universal and generic Whittaker $\mathcal{G}$-modules
(resp. $\widetilde{\mathcal{G}}$-modules) of type $\phi$, where
$\phi:\mathcal{G}^{+}=\widetilde{\mathcal{G}}^{+}\longrightarrow\mathbb{C}$ is
a Lie algebra homomorphism. We classify the isomorphism classes of universal
and generic Whittaker modules. Moreover, we show that a generic Whittaker
modules of type $\phi$ is irreducible if and only if $\phi$ is nonsingular. For
the nonsingular case, we completely determine the Whittaker vectors in
universal and generic Whittaker modules. For the singular case, we concretely
construct some proper submodules of generic Whittaker modules.
