(i) The potential energy (PE) loss when the ball falls 5 m is given by:

\(\text{PE loss} = 0.4g \times 5 = 20 \text{ J}\)

The initial kinetic energy (KE) when the ball starts moving upwards is:

\(\text{Initial KE}_{\text{up}} = 0.4g \times 5 - 12.8 = 7.2 \text{ J}\)

Using the energy conservation principle, the height \(h\) reached is given by:

\(0.4gh = 7.2\)

Solving for \(h\):

\(h = \frac{7.2}{0.4g} = 1.8 \text{ m}\)

(ii) The time taken to fall 5 m is calculated using:

\(5 = 0 + \frac{1}{2} g t_{\text{down}}^2\)

\(t_{\text{down}} = 1 \text{ s}\)

The time taken to reach the greatest height is:

\(0 = 6 - gt_{\text{up}}\) or \(1.8 = \frac{1}{2} g t_{\text{up}}^2\)

\(t_{\text{up}} = 0.6 \text{ s}\)

Total time is:

\(t_{\text{total}} = t_{\text{down}} + t_{\text{up}} = 1 + 0.6 = 1.6 \text{ s}\)