(i) Using the equation of motion: \(s = ut + \frac{1}{2}at^2\).

For PQ: \(s_{PQ} = 20 \times 10 - 0.5 \times a \times 10^2\).

For QR: \(s_{QR} = 20 \times 10 + 0.5 \times a \times 10^2\).

Given \(s_{QR} = 1.5 \times s_{PQ}\), we have:

\(1.5(200 - 50a) = 200 + 50a\).

Simplifying gives \(100 = 125a\) leading to \(a = 0.8 \text{ m s}^{-2}\).

(ii) For distance QS, use \(s = ut + \frac{1}{2}at^2\).

\(s_{QS} = 20 \times 20 + \frac{1}{2} \times 0.8 \times 20^2\).

\(s_{QS} = 400 + 160 = 560 \text{ m}\).

Average speed between Q and S is \(\frac{560}{20} = 28 \text{ m s}^{-1}\).