(i) For a smooth surface, apply Newton's second law to the system:

For A: \(T = 0.8a\)

For B: \(2 - T = 0.2a\)

Combine: \(0.2g = (0.2 + 0.8)a\)

Solve for \(a\): \(a = 2 \, \text{ms}^{-2}\)

Use the equation \(s = \frac{1}{2} a t^2\) to find \(t\):

\(2.5 = \frac{1}{2} \times 2 \times t^2\)

\(t = 1.58 \text{ s}\)

(ii) For a rough surface, the normal force \(N = 8\) and friction \(F = 0.1 \times N = 0.8\).

Apply Newton's second law:

For A: \(T - 0.8 = 0.8a\)

For B: \(2 - T = 0.2a\)

Combine: \(0.2g - 0.8 = (0.2 + 0.8)a\)

Solve for \(a\): \(a = 1.2 \, \text{ms}^{-2}\)

Use the equation \(v^2 = u^2 + 2as\) to find \(v\):

\(v^2 = 0 + 2 \times 1.2 \times 2.5\)

\(v = \sqrt{6} = 2.45 \text{ ms}^{-1}\)