2. Given $P$ is the center of the upper base $\triangle A_{1} B_{1} C_{1}$ of a regular triangular prism $A B C-A_{1} B_{1} C_{1}$, construct a plane $B C D \perp A P$, intersecting the edge $A A_{1}$ at point $D$. If $A A_{1}=2 A B=2$, then the volume of the tetrahedron $D-A B C$ is ( ).
(A) $\frac{\sqrt{3}}{48}$
(B) $\frac{\sqrt{3}}{24}$
(C) $\frac{\sqrt{3}}{16}$
(D) $\frac{\sqrt{3}}{12}$
2. A.

As shown in Figure 3, let the plane $A A_{1} P$ intersect the edges $B C$ and $B_{1} C_{1}$ at points $E$ and $E_{1}$, respectively.
In the rectangle $A E E_{1} A_{1}$, $A A_{1}=2$,
$A E=A_{1} E_{1}=\frac{\sqrt{3}}{2}$,
$$
A_{1} P=\frac{\sqrt{3}}{3} \text {. }
$$

From $\frac{A A_{1}}{A E}=\frac{A P}{E D}$, we get