A

To find the time taken to cover the first height of \( h/4 \) during free fall, we can use the kinematic equations for uniformly accelerated motion.
\( h = \frac{1}{2} g t^2 \)
We know that:
- Total height \( h \)
- Total time to fall \( t \)
Thus, the time taken to fall \( h/4 \) can be derived from the equation:
\( \frac{h}{4} = \frac{1}{2} g t_1^2 \) where \( t_1 \) is the time taken to fall height \( h/4 \).
Since \( t = \sqrt{\frac{2h}{g}} \), we have:
\( t_1 = t \sqrt{\frac{1}{4}} = \frac{t}{2} \).
Now substituting this back to find: \( t_1 = \frac{t}{2} \).
Therefore, the time to cover the first height is \( \frac{t}{2} \).
Hence, the correct answer is Option A: \( \frac{t}{2} \).