1. Rolling without slipping: relation between translation and rotation

When a disc rolls without slipping, the distance travelled by its center is $s = R\theta$ where $\theta$ is the angle rotated.

For half rotation, $\theta = \pi$ so the center moves horizontally by $s = R\pi = \pi R.$

2. Initial position of point A

At $t=0$, point $A$ is the topmost point of the disc.

Let the initial center of the disc be at $C_0=(0,R).$ Then the initial position of the topmost point is $A_0=(0,2R).$

3. Position after half rotation

After half a rotation, the center has moved to the right by $\pi R$. So the new center is $C_1=(\pi R, R).$

A point that was initially at the top of the disc will, after rotating by $\pi$, come to the bottommost point relative to the center. Hence its final position is $A_1=(\pi R, 0).$

4. Displacement of point A

Displacement vector: $\Delta \vec r = A_1 - A_0 = (\pi R - 0,\; 0-2R) = (\pi R, -2R).$

Magnitude of displacement: $|\Delta \vec r| = \sqrt{(\pi R)^2 + (-2R)^2}$ $= R\sqrt{\pi^2+4}.$

5. Match with options

This corresponds to: $\boxed{R\sqrt{\pi^2+4}}$ which is Option C.

6. Comparison with stored answer

Stored correct answer: C

Our derived answer: C

So they agree.