2. In tetrahedron $ABCD$, let $AB=1, CD=$ $\sqrt{3}$, the distance between line $AB$ and line $CD$ is 2, and the angle between them is $\frac{\pi}{3}$. Then the volume of tetrahedron $ABCD$ is ( ).
(A) $\frac{\sqrt{3}}{2}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) $\frac{\sqrt{3}}{3}$
2.C.

Through point $D$, draw $D F \perp C B$, through point $A$, draw $A E \perp B C$, connect $C E$, $E D$, $A F$, $B F$, and complete the pyramid into a prism. Therefore, the volume of the pyramid is $\frac{1}{3} \times \frac{1}{2} C E \cdot C D \sin \angle E C D \cdot h=\frac{1}{2}$.