(i) To find the time for which P's height is greater than 15 m, use the equation for vertical motion:

\(h = ut - \frac{1}{2}gt^2\)

where \(u = 20\) m s^-1 and \(g = 10\) m s^-2. Set \(h = 15\) m:

\(15 = 20t - 5t^2\)

Solve the quadratic equation:

\(5t^2 - 20t + 15 = 0\)

Divide by 5:

\(t^2 - 4t + 3 = 0\)

Factorize:

\((t - 1)(t - 3) = 0\)

So, \(t = 1\) or \(t = 3\). The duration is \(3 - 1 = 2\) seconds.

(ii) To find the velocities when P and Q are at the same height, equate their height equations:

\(20t - 5t^2 = 25(t - 0.4) - 5(t - 0.4)^2\)

Expand and simplify:

\(20t - 5t^2 = 25t - 10 - 5(t^2 - 0.8t + 0.16)\)

\(20t - 5t^2 = 25t - 10 - 5t^2 + 4t - 0.8\)

Combine terms:

\(20t = 29t - 10.8\)

\(9t = 10.8\)

\(t = 1.2\) (or \(t = 0.8\))

Use \(v = u - gt\) to find velocities:

For P:

\(v_P = 20 - 10 \times 1.2 = 8 \text{ m s}^{-1}\)

For Q:

\(v_Q = 25 - 10 \times (1.2 - 0.4) = 17 \text{ m s}^{-1}\)