To solve this problem, we use the equations of motion under constant acceleration due to gravity, where the acceleration is \(g = 9.8 \text{ m/s}^2\).

(i) Speed at the mid-point of OA:

The maximum height A is found using the equation: \(0 = u^2 - 2gs\), where \(u = 8 \text{ m/s}\) and \(v = 0 \text{ m/s}\) at the highest point.

\(0 = 8^2 - 2g \cdot s\)

\(s = \frac{64}{2 \times 9.8} = 3.2 \text{ m}\)

The mid-point of OA is at \(s = 1.6 \text{ m}\).

Using \(v^2 = u^2 - 2gs\) to find the speed at the mid-point:

\(v^2 = 8^2 - 2 \times 9.8 \times 1.6\)

\(v^2 = 64 - 31.36\)

\(v = \sqrt{32.64} \approx 5.66 \text{ m/s}\)

(ii) Time to reach the mid-point of OA:

Using \(v = u - gt\), where \(v = 5.66 \text{ m/s}\) at the mid-point:

\(5.66 = 8 - 9.8t\)

\(9.8t = 8 - 5.66\)

\(t = \frac{2.34}{9.8} \approx 0.234 \text{ s}\)