Step 1: Geometric reading:
$\bar a\times(\bar a\times\bar c)$ equals $-\bar c_{\perp}$, minus the part of $\bar c$ perpendicular to $\bar a$ (since $\bar a$ is a unit vector). So $\bar b = \bar c_\perp$.
Step 2: Magnitudes:
$|\bar c_\perp| = |\bar c|\sin\theta = 2\sin\theta$. Given $|\bar b| = 1$, $\sin\theta = \frac12$.
Step 3: Angle:
The acute angle with $\sin\theta = \frac12$ is $\theta = \frac\pi6$.
Step 4: Check with option (D):
$\cos\frac\pi6 = \frac{\sqrt3}2$, so $\bar a\cdot\bar c = \sqrt3$ and $t^2 = 3$, matching the algebraic result.
Final Answer:
Option (D).
\[ \boxed{\frac{\pi}{6} \text{ (D)}} \]